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Complete Unit 4

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Table of Contents

Unit 4 Pacing Chart -------------------------------------------------------------------------------------------- 1

Geometry Unit 4 Skills List ---------------------------------------------------------------------------------------- 5

Unit 4 Lesson Plans -------------------------------------------------------------------------------------------- 6

Day 46 Bellringer -------------------------------------------------------------------------------------------- 33

Day 46 Activity -------------------------------------------------------------------------------------------- 36

Day 46 Practice -------------------------------------------------------------------------------------------- 38

Day 46 Exit Slip -------------------------------------------------------------------------------------------- 42

Day 47 Bellringer -------------------------------------------------------------------------------------------- 44

Day 47 Activity -------------------------------------------------------------------------------------------- 46

Day 47 Practice -------------------------------------------------------------------------------------------- 48

Day 47 Exit Slip -------------------------------------------------------------------------------------------- 54

Day 48 Bellringer -------------------------------------------------------------------------------------------- 56

Day 48 Activity -------------------------------------------------------------------------------------------- 58

Day 48 Practice -------------------------------------------------------------------------------------------- 61

Day 48 Exit Slip -------------------------------------------------------------------------------------------- 67

Day 49 Bellringer -------------------------------------------------------------------------------------------- 69

Day 49 Activity -------------------------------------------------------------------------------------------- 71

Day 49 Practice -------------------------------------------------------------------------------------------- 73

Day 49 Exit Slip -------------------------------------------------------------------------------------------- 78

Week 10 Assessment -------------------------------------------------------------------------------------------- 80

Day 51 Bellringer -------------------------------------------------------------------------------------------- 86

Day 51 Activity -------------------------------------------------------------------------------------------- 88

Day 51 Practice -------------------------------------------------------------------------------------------- 91

Day 51 Exit Slip -------------------------------------------------------------------------------------------- 96

Day 52 Bellringer -------------------------------------------------------------------------------------------- 98

Day 52 Activity -------------------------------------------------------------------------------------------- 101

Day 52 Practice -------------------------------------------------------------------------------------------- 104

Day 52 Exit Slip -------------------------------------------------------------------------------------------- 111

Day 53 Bellringer -------------------------------------------------------------------------------------------- 113

Day 53 Activity -------------------------------------------------------------------------------------------- 116

Day 53 Practice -------------------------------------------------------------------------------------------- 118

Day 53 Exit Slip -------------------------------------------------------------------------------------------- 120

Day 54 Bellringer -------------------------------------------------------------------------------------------- 123

Day 54 Activity -------------------------------------------------------------------------------------------- 125

Day 54 Practice -------------------------------------------------------------------------------------------- 129

Day 54 Exit Slip -------------------------------------------------------------------------------------------- 134

Week 11 Assessment -------------------------------------------------------------------------------------------- 136

Day 56 Bellringer -------------------------------------------------------------------------------------------- 145

Day 56 Activity -------------------------------------------------------------------------------------------- 147

Day 56 Practice -------------------------------------------------------------------------------------------- 149

Day 56 Exit Slip -------------------------------------------------------------------------------------------- 157

Day 57 Bellringer -------------------------------------------------------------------------------------------- 159

Day 57 Activity -------------------------------------------------------------------------------------------- 161

Day 57 Practice -------------------------------------------------------------------------------------------- 163

Day 57 Exit Slip -------------------------------------------------------------------------------------------- 167

Day 58 Bellringer -------------------------------------------------------------------------------------------- 169

Day 58 Activity -------------------------------------------------------------------------------------------- 172

Day 58 Practice -------------------------------------------------------------------------------------------- 174

Day 58 Exit Slip -------------------------------------------------------------------------------------------- 179

Day 59 Bellringer -------------------------------------------------------------------------------------------- 181

Day 59 Activity -------------------------------------------------------------------------------------------- 184

Day 59 Practice -------------------------------------------------------------------------------------------- 187

Day 59 Exit Slip -------------------------------------------------------------------------------------------- 195

Week 12 Assessment -------------------------------------------------------------------------------------------- 197

Day 61 Bellringer -------------------------------------------------------------------------------------------- 204

Day 61 Activity -------------------------------------------------------------------------------------------- 207

Day 61 Practice -------------------------------------------------------------------------------------------- 209

Day 61 Exit Slip -------------------------------------------------------------------------------------------- 215

Day 62 Bellringer -------------------------------------------------------------------------------------------- 217

Day 62 Activity -------------------------------------------------------------------------------------------- 219

Day 62 Practice -------------------------------------------------------------------------------------------- 222

Day 62 Exit Slip -------------------------------------------------------------------------------------------- 227

Day 63 Bellringer -------------------------------------------------------------------------------------------- 229

Day 63 Activity -------------------------------------------------------------------------------------------- 231

Day 63 Practice -------------------------------------------------------------------------------------------- 233

Day 63 Exit Slip -------------------------------------------------------------------------------------------- 237

Day 64 Bellringer -------------------------------------------------------------------------------------------- 239

Day 64 Activity -------------------------------------------------------------------------------------------- 242

Day 64 Practice -------------------------------------------------------------------------------------------- 244

Day 64 Exit Slip -------------------------------------------------------------------------------------------- 248

Unit 4 Test -------------------------------------------------------------------------------------------- 250

Unit 4 Pacing Chart

HighSchoolMathTeachers.com ©2020 Page 1

Unit Week Day CCSS Standards Objective I Can Statements

Unit 4 Triangle

Congruence

Week 10 – Transformat

ions to Theorems

46

CCSS.MATH.CONTENT.HSG.CO.B.6 Use geometric descriptions of rigid motions to

transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use

the definition of congruence in terms of rigid motions to decide if they are congruent.

Come up with a result(conclusion) as a

result : Any rigid motion preserves

angle measure. Any rigid motion of the

plane is a reflection, rotation, translation or a

glide reflection. Any rigid motion maps straight segments to

straight segments, lines to lines, and circles to circles. An rigid motion maps any three non-collinear points into non-collinear points.

I can explain how and why: Any rigid motion preserves angle

measure. Any rigid motion of the plane is a reflection, rotation, translation or

a glide reflection. Any rigid motion maps straight segments to straight segments,

lines to lines, and circles to circles.

An rigid motion maps any three non-collinear points into non-

collinear points.

Unit 4 Triangle

Congruence


ions to Theorems

47




Come up with a result(conclusion) as a

result: Any plane rigid motion is

invertible. Any rigid motion with a

fixed point is either a reflection or a rotation

I can explain how and why: Any plane rigid motion is

invertible. Any rigid motion with a fixed

point is either a reflection or a rotation

Unit 4 Triangle

Congruence


ions to Theorems

48




Come up with a conclusion as a result:

The composition of two rigid motions is also a rigid

motion The composition of a half-turn and a reflection is a

glide reflection.

I can explain howand why: The composition of two rigid motions is also a rigid motion The composition of a half-turn

and a reflection is a glide reflection.

Unit 4 Pacing Chart


Unit 4 Triangle

Congruence


ions to Theorems

49




Come up with a conclusion as a result:

Successive reflection in two intersecting mirror lines

produces a rotation about the point of intersection through twice the angle

between the mirror lines. Successive reflection in

parallel mirror lines produces a translation in a direction perpendicular to

the mirrors through a distance equal to twice the

distance between the mirrors

Any rigid motion of the Euclidean plane can be

written as a composition of no more than 3 reflections.

I can explain how and why; Successive reflection in two

intersecting mirror lines produces a rotation about the point of

intersection through twice the angle between the mirror lines. Successive reflection in parallel

mirror lines produces a translation in a direction

perpendicular to the mirrors through a distance equal to twice the distance between the mirrors

and

Any rigid motion of the Euclidean

plane can be written as a composition of no more than 3

reflections.

Unit 4 Triangle

Congruence


ions to Theorems

50 Assessment Assessment Assessment

Unit 4 Triangle

Congruence

Week 11 – Proofs of

Congruent Triangles

51

CCSS.MATH.CONTENT.HSG.CO.B.7 Use the definition of congruence in terms of rigid

motions to show that two triangles are congruent if and only if corresponding pairs of sides and

corresponding pairs of angles are congruent.

Identifying Congruent sides of a triangle

I can Identify congruent sides of a triangle

Unit 4 Pacing Chart


Unit 4 Triangle

Congruence


Congruent Triangles

52




Identifying Congruent angles of a triangle

I can Identifying congruent angles of a triangle

Unit 4 Triangle

Congruence


Congruent Triangles

53




Use the definition of congruence in terms of

rigid motions to show that two triangles are congruent if and only if corresponding

pairs of sides and corresponding pairs of angles are congruent.

I can use the definition of congruence in terms of rigid

motions to show that two triangles are congruent if and only if corresponding pairs of

sides and corresponding pairs of angles are congruent.

Unit 4 Triangle

Congruence


Congruent Triangles

54




Show Congruence of image and object distance in

dilation if scale factor is -1.

I can show congrence of image and object distance in dilation if

the scale factor is -1.

Unit 4 Triangle

Congruence


Congruent Triangles


Unit 4 Triangle

Congruence

Week 12 – Congruent

Parts of Congruent Figures are Congruent

56

CCSS.MATH.CONTENT.HSG.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence

in terms of rigid motions.

Use the postulate SSS to show that triangles are

congruent. Also apply it so solve other geometric

problems.

I can use the postulate SSS to show that triangles are

congruent. I can apply the postulate so solve

other geometric problems.

Unit 4 Pacing Chart


Unit 4 Triangle

Congruence



57



Use the postulate ASA to show that triangles are


problems.

Use the postulate ASA to show that triangles are congruent. Also apply it so solve other geometric

problems.

Unit 4 Triangle

Congruence



58



Use the postulate SAS to show that triangles are


problems.

I can use the postulate SAS to show that triangles are

congruent. I can apply the postulate to solve


Unit 4 Triangle

Congruence



59



Use the postulates SSS, SAS and ASA to show that triangles are congruent. Also apply them to solve


I can use the postulates SSS, SAS and ASA to show that triangles are congruent and also apply

them to solve other geometric problems.

Unit 4 Triangle

Congruence




Geometry Unit 4 Skills List

HighSchoolMathTeachers.com©2020 Page 5

Geometry Unit 4 Skills List

Number Unit CCSS Skill

15 4 HSG.CO.B.6 Describing transformation theorems

16 4 HSG.CO.B.7 Showing congruence in triangles

17 4 HSG.CO.B.8 Explain the criteria for triangle

congruence

Unit 4 Lesson Plan


Unit: Unit 4 Triangle Congruence

Course: Geometry

Topic: Week 10 – Transformations to Theorems

Day: 46

Common Core State Standard: CCSS.MATH.CONTENT.HSG.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Mathematical Practice: Assessment

Objective: Come up with a result(conclusion) as a result : Any rigid motion preserves angle measure. Any rigid motion of the plane is a reflection, rotation, translation or a glide reflection. Any rigid motion maps straight segments to straight segments, lines to lines, and circles to circles. An rigid motion maps any three non-collinear points into non-collinear points.

I can statement: I can explain how and why: Any rigid motion preserves angle measure. Any rigid motion of the plane is a reflection, rotation, translation or a glide reflection. Any rigid motion maps straight segments to straight segments, lines to lines, and circles to circles. An rigid motion maps any three non-collinear points into non-collinear points.

Unit 4 Lesson Plan


Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will reflect different shapes with a plane mirror and compare the geometric shape of the image with that of the object 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.

Materials: Bellringer 46 Day 46 Activities Day 46 Practice Day 46 Presentation Day 46 Exit Slip

Accommodations/Special Circumstances:

Technology:

Reflection:

Extra/Additional Resources:

Unit 4 Lesson Plan



Course: Geometry


Day: 47


Mathematical Practice: CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP5 Use apprpriate tools strategically. CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Objective: Come up with a result(conclusion) as a result: Any plane rigid motion is invertible. Any rigid motion with a fixed point is either a reflection or a rotation

I can statement: I can explain how and why: Any plane rigid motion is invertible. Any rigid motion with a fixed point is either a reflection or a rotation

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 4. They will illustrate the fixed points of an pre-images under rotation 3. The presentation will be used to look for misconceptions and encourage discussion.


Unit 4 Lesson Plan


4. Students will complete the exit slip before leaving for the day.


Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 48



Objective: Come up with a conclusion as a result: The composition of two rigid motions is also a rigid motion The composition of a half-turn and a reflection is a glide reflection.

I can statement: I can explain howand why: The composition of two rigid motions is also a rigid motion The composition of a half-turn and a reflection is a glide reflection.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of three to discover that the composition of two rigid motions is also a rigid motion


Unit 4 Lesson Plan


3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 49



Objective: Come up with a conclusion as a result: Successive reflection in two intersecting mirror lines produces a rotation about the point of intersection through twice the angle between the mirror lines. Successive reflection in parallel mirror lines produces a translation in a direction perpendicular to the mirrors through a distance equal to twice the distance between the mirrors Any rigid motion of the Euclidean plane can be written as a composition of no more than 3 reflections.

I can statement: I can explain how and why; Successive reflection in two intersecting mirror lines produces a rotation about the point of intersection through twice the angle between the mirror lines. Successive reflection in parallel mirror lines produces a translation in a direction perpendicular to the mirrors through a distance equal to twice the distance between the mirrors

Unit 4 Lesson Plan


and Any rigid motion of the Euclidean plane can be written as a composition of no more than 3 reflections.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 5. They will view image of an object between two parallel plane mirrors to discover their features 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 50

Common Core State Standard: Assessment


Objective: Assessment

I can statement: Assessment

Procedures: Assessment

Materials: Assessment


Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry

Topic: Week 11 – Proofs of Congruent Triangles

Day: 51

Common Core State Standard: CCSS.MATH.CONTENT.HSG.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quatitatively.

Objective: Identifying Congruent sides of a triangle

I can statement: I can Identify congruent sides of a triangle

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 5 to illustrate the idea of congruent sides using rigid motions of a simply constructed triangle 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:

Unit 4 Lesson Plan


Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 52



Objective: Identifying Congruent angles of a triangle

I can statement: I can Identifying congruent angles of a triangle

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of four to identify congruent angles in congruent triangles 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:

Unit 4 Lesson Plan


Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 53


Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quatitatively. CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

Objective: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

I can statement: I can use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 5. They like show that if two triangles are congruent, then they have corresponding angles and sides. 3. The presentation will be used to look for misconceptions and encourage discussion.


Unit 4 Lesson Plan


4. Students will complete the exit slip before leaving for the day.


Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 54


Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

Objective: Show Congrence of image and object distance in dilation if scale factor is -1.

I can statement: I can show congrence of image and object distance in dilation if the scale factor is -1.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will draw a triangle and its image under a dilation with a scale factor of -1 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Unit 4 Lesson Plan



Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 55








Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry

Topic: Week 13 – Rigid Motion – Putting it all together

Day: 61


Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quatitatively. CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

Objective: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

I can statement: I can use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure Given two figures, I can use the definition of congruence in terms of rigid motions to decide if they are congruent.

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. They will cut different triangles out of a rectangular plane paper and identify congruent and non congruent triangles


Unit 4 Lesson Plan


3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 62



Objective: Identify Conguent angles and sides

I can statement: I can Identify Conguent angles and sides

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 2. They will learn how to identify congruent angles and sides using congruent triangles 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.



Technology:

Unit 4 Lesson Plan


Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 63



Objective: Show that two triangles are or are not congruent using the biconditional statement in the standard

I can statement: I can show that two triangles are or are not congruent using the biconditional statement in the standard

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3. Students will establish that if corresponding angles and sides of a triangle congruent, then the two triangles are congruent. 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Unit 4 Lesson Plan



Technology:

Reflection:


Unit 4 Lesson Plan



Course: Geometry


Day: 64

Common Core State Standard: CCSS.MATH.CONTENT.HSG.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.


Objective: Show that two triangles are or are not congruent using the postulates

I can statement: I can show that two triangles are or are not congruent using the postulates

Procedures: 1. Students will complete the bellringer. 2. Students will work in groups of at least 3.They will identify congruent triangles in a square that is divided into triangles by the diagonals 3. The presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete the exit slip before leaving for the day.


Unit 4 Lesson Plan



Technology:

Reflection:


Day 46 Bellringer



Course: Geometry


Day: 65








Technology:

Reflection:


Day 46 Bellringer


Identify the rigid motion involved in each of the following transformations.

a)

b)

c)

Day 46 Bellringer


2. Which three rigid motions can be used to map the squares below?

3. The diagram below shows a square and a mirror line. Sketch the image of the square

Day 46 Bellringer


Answer Key Day 46

1. a) Glide reflection

b) Translation

c) reflection

2. Reflection

Translation

Rotation

3.

Day 46 Activity


1. Draw a line of 2 in on a plane paper.

2. Hold a plane mirror vertically and close to the line such that the image of the line is formed by the

plane mirror.

What is the geometric shape of the image?

3. Draw a circle of a convenient radius on a plane paper.

4. Hold the mirror vertically and close to the circle.

What is the geometric figure of the image?

5. Hold the mirror in different positions around the circle and observe whether shape of the image

changes.

Does the shape of the image change?

Day 46 Activity


In this activity students will reflect different shapes with a plane mirror and compare the geometric

shape of the image with that of the object. Students will work in groups of at least three and each group

is required to have a plane mirror, a plane paper, a pencil, a ruler and a compass.

Answer Keys

Day 46:

1. No response

2. line

3. No response

4. Circle

5. No

Day 46 Practice


Use the following information to answer questions 1 and 2.

A square of sides 6in by 6in was translated two units upwards and 3 units to the left.

1. What was the geometric figure of the image.

2. What was the area of the image.

Use the following information to answer questions 3 and 4.

A line of length 5 in is reflected through a line below it.

3. What is the shape of the image?

4. What is the length of the image?

In questions 5 to 12, State whether the given statement is true or false.

5. Rigid motions map a square to another smaller square.

6. A reflection maps a triangle to another triangle of the same size.

7. A rotation maps a right angle triangle to a scalene triangle.

8. A translation maps a square to a rectangle.

9. Any transformation that maps an object to an image of the same size and shape as the object is a rigid

motion.

10. A rotation can map non-collinear points to collinear points.

11. A reflection maps two intersecting lines into another two intersecting lines with different angle of

intersection.

12. Any rigid motion will map collinear points to collinear points.

Day 46 Practice


Use the figure below to answer questions 13 – 15.

13. What is the value of ∝?

14. What is the value of 𝜃?

15. Find the value of 𝜑

67°

∝ 𝜃

𝜑

Day 46 Practice


Use the figure below to answer questions 16 to 19. (figures are not drawn to scale)

16. Which rigid motion can map the two triangles?

17. What is the value of ∅?

18. What is the value of 𝜃?

19. What is the length of the side labelled y?

20. ∆𝐴𝐵𝐶 is reflected to map onto ∆𝐴′𝐵′𝐶′. If the side AB is 2 in long, What is the length of the side

𝐴′𝐵′ in the image.

44°

5in

y

∅

𝜃

Day 46 Practice


Answer Keys Day 46:

1. square

2. 36𝑖𝑛2

3. Line

4. 5 𝑖𝑛

5. False

6. True

7. False

8. False

9. True

10. False

11. False

12. True

13. 67°

14. 67°

15. 46°

16. Translation

17. 44°

18. 46°

19. 5in

20. 2in

Day 46 Exit Slip


1. A right triangle with sides 3 in, 4 in and 5 in is rotated in anti-clockwise through an angle of 90°. What

is the length of the longest side in the image.

Day 46 Exit Slip


Answer Keys

Day 46:

1. 5 in

Day 47 Bellringer


1. A triangle of area 15𝑠𝑞. 𝑖𝑛 is reflected twice about different mirror lines. Find the area of the final

image.

2. Explain your answer in 1 above.

3. A square of rotated through 3 full turns about the origin. What would be the interior angles of the

final image?


5. Draw the image of the following lines after reflection about the given mirror line.

Day 47 Bellringer


Answer Key

Day 47

1. 15 𝑠𝑞. 𝑖𝑛

2. The size of the triangle does not change under reflection(s)

3. All angles are 90°

4. The shape, consequently, the angle measure of a figure does not change due to rotation(s).

5.

Day 47 Activity


1. Mount the plane paper on the softwood using the masking tape.

2. Place the plastic bar on the peace of paper and draw its pre-image by making a line along its

boundaries.

3. Select any point on the bar and drive the pushpin so that it holds the bar against the paper on the

softwood.

4. Using one of your fingers, push one side (unmounted side) of the bar so that it rotates due to the

force.

5. Stop when the bar makes a rotation between 90° and 180°.

6. Trace the image of the bar at that particular point and label it the pre-image. Remove the bar and the

pin from the plane paper.

7. Which direction of rotation did you take?

8. Which angle did the bar rotate, positive or negative?

9. Identify the name given to the position of the pin in relation to the motion of the bar.

10. By keenly looking at the image and its pre-image, which point did not move?

11. Can you make a conclusion about fixed points in relation to the motion of the bar?

Day 47 Activity


In this activity, we would like to illustrate the fixed points of an pre-images under rotation. Each group is

required to have a pushpin, a thin plastic short bar (about 3 in – 4 in) or a thin wooden bar that can be

placed through by the pushpin, a plane A4 paper, masking tape, softwood and a pencil.

Answer Keys

Day 47:

1 – 6.No response

7. Difference responses; clockwise or counterclockwise

8. If clockwise the answer should be negative and if counterclockwise the answer is positive

9. Center of rotation

10. Point at the position of the pin

11. In rotation, the points of the pre-image that is also the center of rotation remains fixed after rotation

of any angle.

Day 47 Practice


Use the following information to answer questions 1 - 10.

In each of the following identify the transformation that will map the image back to the pre image given

the pre-image and the initial transformation.

1. Image of a reflection of a triangle along its smallest side.

2. Image of a reflection of a rectangle about one of its diagonal.

3. Image after a rotation of 60° about the origin.

4. Image after translation of vector (2

−5).

5. Image after translation of vector (−3−1

).

6. Imge after rotating a pentagon through a negative quarter turn about (1,0).

7. Image after translating an object 5 units upwards and 3 units to the right.

8. Image after rotating a triangle 187° in the counterclockwise direction.

9. Image after reflection an equilateral triangle about 𝑥 = −3.

10. Image of a unit square after reflecting it about 𝑦 + 𝑥 = 0.

Use the following statement to answer questions 11 – 17.

Identify the fixed points of the pre-image after the following motion

11. Reflection about line M

M

Day 47 Practice


12. Rotation about O.

13. 360° rotation of a unit circle about point (−5,7).

14. Reflection of a right triangle along its hypotenuse.

15. The following translation

16.

O

A

B

C

E

C’

D’ A’

B’

D E’

P Q

R

𝑇

𝑉

Day 47 Practice


17.

Use the following information to answer questions 18 – 20.

Find the pre-image given the following images after the transformations given.

18. Reflection about the vertical line

M

V

P

Q

L

Day 47 Practice


19. Rotated through a negative quarter turn about the center.

20. Translated 3 units up and 1 unit to the right

Day 47 Practice


Answer Keys

Day 47

1. Reflect the image about the same smallest side

2. Reflect the rectangle about the same diagonal

3. Rotate the image through −60° about the origin

4. Translate the image through vector (−25

).

5. Translate the image through vector (31

).

6. Rotate the image (the pentagon) through positive quarter turn about (1,0).

7. Translate the object 5 units downwards and 3 units to the left.

8. Rotate the triangle through −187° .

9. Reflect the equilateral triangle about 𝑥 = −3.

10. Reflect the unit square about 𝑦 + 𝑥 = 0.

11. Points on the pre-image that lies on line M

12. Point Q

13. All points on the circle

14. Points along the hypotenuse

15. None

16. Point Q

17. Points along line PQ

18.

19.

Day 47 Practice


20.

Day 47 Exit Slip


Identify the fixed point(s) of rectangle rotated about 180° about a point that lies on one of its diagonal.

Day 47 Exit Slip


Answer Keys

Day 47

The fixed point is the center of rotation which is a point on one of its diagonals

Day 48 Bellringer


1. (a) List any three types of rigid motions of the plane you have learnt in class.

(b) What happens to the shape of the pre-image after undergoing a given rigid motion?

(c) What happens to the size of the pre-image after undergoing a given rigid motion?

(d) What happens to the angle measures of the pre-image after undergoing a given rigid motion to form

the corresponding image?

2. (a) If a rectangle is rotated about a given point along the coordinate plane, then the image further

reflected about a given line of reflection, what happens to its shape and size after the two transformations?

(b) The figure below represents a pre-image and its corresponding image after a transformation.

Is this kind of transformation a rigid motion or not? Give a reason to support your answer.

Pre-image

Image

Day 48 Bellringer


Answer Keys

Day 48:

1. (a) Rotations, reflections, translations and glide reflections.

(b) The shape remains unchanged/ it is preserved

(c) The size remains unaltered/ it is preserved

(d) Angle measure is preserved

2. (a) The size and shape are both preserved after the rigid motions

(b) It is not a rigid motion, though the shape is preserved, the size has changed after the

transformation.

Day 48 Activity


1. Draw the coordinate plane on the grid using a ruler labelling the x-axis and the y-axis as shown below.

2. On the grid, draw triangle PQR in the position as shown below. This is the pre-image.

Day 48 Activity


3. Translate the triangle through (𝑥 + 2, 𝑦) to its image along the coordinate plane and label it P′Q′R′.

4. Examine the two triangles, measure their corresponding lengths using a ruler and their corresponding

angles using a protractor. What does this tell you about their shape and size?

5. Now, using triangle P′Q′R′ as your pre-image reflect it along the x-axis and label the resulting image

P′′Q′′R′′.

6. Consider triangles P′Q′R′ and P′′Q′′R′′. Measure their corresponding lengths using a ruler and their

corresponding angles using a protractor. What does this tell you about their shape and size?

7. Consider triangles PQR and P′′Q′′R′′. Measure their corresponding lengths using a ruler and their

corresponding angles using a protractor. What does this tell you about the shape and size?

8. What do you conclude about the three triangles after performing the two transformations? Base your

conclusion on the responses you have given in questions 6 and 7.

Day 48 Activity


In this activity students will work in groups of three to discover that the composition of two rigid

motions is also a rigid motion. The students in the respective groups will require a graph paper (grid), a

ruler and a protractor.

Answer Keys Day 48:

1. No response

2. No response

3. No response

4. They have the same size and shape/ They are congruent

5. No response



8. All the three triangles are congruent/ Shape and size is preserved.

Day 48 Practice


Study the diagram below and use it to answer questions 1- 8.

1. What type of transformation maps Δ ABC onto Δ A′B′C′?

2. What type of transformation maps Δ A′B′C′ onto Δ A′′B′′C′′?

3. Considering Δ ABC and Δ A′′B′′C′′ show the mapping of corresponding line segments in the two

triangles.

4. Considering Δ ABC and Δ A′′B′′C′′ show the mapping of corresponding angles in the two triangles.

5. Basing on questions 3 and 4 above, what do you say about Δ ABC and Δ A′′B′′C′′?

6. Identify one type of rigid motion that maps Δ ABC and Δ A′′B′′C′′.

A

B C

B′ C′

A′

B′′ C′′

A′′

Mirror line, l

Day 48 Practice


7. Considering the two types of rigid motions shown above, does Δ ABC change its shape and size after

being transformed onto the final image Δ A′′B′′C′′?

8. According to question 7 above, what is your conclusion about the composition of two rigid motions?

The diagram below shows a sequence of two rigid motions that map rectangle WXYZ onto the final

image. Study it and use it to answer questions 9-14.

9. What type of rigid motion maps rectangle WXYZ onto rectangle W′X′Y′Z′?

10. What type of rigid motion maps rectangle W′X′Y′Z′ onto W′′X′′Y′′Z′′?

11. Considering rectangle WXYZ and rectangle W′′X′′Y′′Z′′ show the mapping of corresponding sides.

12. Considering rectangle WXYZ and rectangle W′′X′′Y′′Z′′ show the mapping of corresponding angles.

13. How are the sizes of rectangles WXYZ and W′′X′′Y′′Z′′ related basing on questions 11 and 12 above?

Day 48 Practice


14. Considering the two transformations and the sizes and shapes of the three rectangles, what do you

discover?

The grid below shows a composition of two rigid motions, study it then answer questions 15-20.

15. Give that ΔPQR is mapped onto ΔP′Q′R′ through a rotation of 180° about the point (0,0), what type

of rigid motion maps Δ P′Q′R′ onto Δ P′′Q′′R′′?

Day 48 Practice


16. On the grid draw the image of Δ P′′Q′′R′′ using dotted lines after a reflection along the x-axis.

17. What type of rigid transformation is supposed to map the dotted triangle you have drawn in 16

above to ΔPQR?

18. Considering ΔPQR, the dotted triangle and Δ P′′Q′′R′′. Which single rigid motion will map

Δ P′′Q′′R′′onto ΔPQR?

19. Considering the type of transformation you have identified in question 18 above, what do you notice

about relationship between the shape and size of ΔPQR and ΔP′′Q′′R′′?

20. Considering the two sequences of transformations listed below, what do you deduce?

Sequence 1: ΔPQR → ΔP′Q′R′ → Δ P′′Q′′R′′

Sequence 2: ΔP′′Q′′R′′ → dotted triangle → Δ PQR

P

Q

y

x

R

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

5

4

3

2

1

−1

−2

P′

Q′ R′ Q′′ R′′

P′′

Day 48 Practice


Answer keys

Day 48:

1. Reflection

2. Translation

3. AB → A′′B′′; BC → B′′C′′; CD → C′′D′′

4. ∠A → ∠A′′, ∠B → ∠B′′, ∠C → ∠C′′

5. ABCD ≅ A′′B′′C′′D′′

6. Glide reflection

7. No

8. The composition of two rigid motions is also a rigid motion

9. Reflection

10. Rotation

11. WX → W′′X′′; XY → X′′Y′′; YZ → Y′′Z′′ and WZ → W′′Z′′

12. ∠W → ∠W′′, ∠X → ∠X′′, ∠Y → ∠Y′′and ∠Z → ∠Z′′

13. WXYZ ≅ W′′X′′Y′′Z′′

14. A sequence of two rigid motions is still a rigid motion, distances and angle measures are preserved.

15. Reflection

Day 48 Practice


16.

17. Translation

18. Glide reflection

19. ΔPQR ≅ Δ P′′Q′′R′′

20. The two sequences of rigid motions are the same. The composition of a half-turn and a reflection is

just a glide reflection.

P

Q

y

x

R

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

5

4

3

2

1

−1

P′

Q′ R′ Q′′ R′′

P′′

Day 48 Exit Slip


The grid below shows a composition of two rigid motions, a half turn about the origin (0,0) followed by

a reflection along the line 𝑥 = 1 performed on a square PQRS.

(a) On the grid draw the image of square P′′Q′′R′′S′′ using dotted lines after a reflection along the x-axis.

(b) What type of transformation will map the dotted square you have drawn in (a) above to square

PQRS?

(c) Consider square PQRS, the dotted square and square P′′Q′′R′′S′′. What single transformation will

map square P′′Q′′R′′S′′ onto square PQRS?

(d) Given that square PQRS has been transformed through a half turn about the origin (0,0) followed by

a reflection along the line 𝑥 = 1, relate this transformation to the transformation described in (c) above.

What do you notice?

P

Q

y

x

R

S

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

5

4

3

2

1

−1

−2

P′

Q′

S′

R′ Q′′ R′′

P′′ S′′

Day 48 Exit Slip


Answer Keys Day 48:

(a)

(b) A translation

(c) A glide reflection

(d) The composition of a half-turn and a reflection is a glide reflection.

P

Q

y

x

R

S

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

5

4

3

2

1

−1

−2

P′

Q′

S′

R′ Q′′ R′′

P′′ S′′

Day 49 Bellringer


1. Use the graph below to answer the questions that follow

a) If ∆𝐴𝐵𝐶 is rotated −180° about the origin, what will be the new coordinates of point C?

b) Indicate the image on the graph above, if ∆𝐴𝐵𝐶 is translated 5 units upwards along y-axis.

c) What will the new coordinates of point B if ∆𝐴𝐵𝐶 is reflected over the line 𝑥 = 1?

d) What will be the new coordinates of point B if ∆𝐴𝐵𝐶 is reflected over the x-axis?

2. The triangle below is reflected over the given mirror line. Draw the image.

Day 49 Bellringer


Answer Key

Day 4

1 a) (1,1)

b)

c) (3, −4)

d) (−1,4)

2.

Day 49 Activity


1. Draw two parallel lines 6 inches apart on a plane paper and label them 𝐿1 and 𝐿2.

2. Write the letter ‘K’ in between the lines as shown.

3. Hold the plane mirror vertically on 𝐿1 and view the letter ‘K’.

Does it have the same orientation as the original letter?

4. Hold the plane mirror vertically on 𝐿2 and view the letter ‘K’.

Does it have the same orientation as the original letter?

5. Position the plane mirrors vertically on both lines and observe the images of letter ‘K’.

6. Observe the second image from each plane mirror.

Does it appear like a translation of the letter ‘K’?

K

𝐿1 𝐿2

Day 49 Activity


In this activity students will view image of an object between two parallel plane mirrors.

Students will work in groups of at least 5 and each group is required to have two plane mirrors, a plane

paper and a pen.

Answer Keys

Day 49:

1-2. No response

3. No

4. No

5. No response

6. Yes

Day 49 Practice


A square measuring 1in by 1in was placed between two plane mirrors 2 inches apart as shown. Use this

diagram to answer questions 1-5.

1. What is least distance we can translate the square to the left such that it coincides with its image

resulting from two successive reflection of the two mirrors?

2. What will be the distance between the image of the square and 𝑀1 if the square is translated to the

left until it coincides with the image resulting from successive reflection of the two mirrors?


left until it coincides with the image resulting from successive reflection of the two mirrors?


right until it coincides with the image resulting from successive reflection of the two mirrors?


right until it coincides with the image resulting from successive reflection of the two mirrors?

Use the information below to answer questions 6 and 7.

A circle centered at the origin is drawn in 𝑥𝑦 plane such that two parallel and vertical mirror lines makes

a tangent with it. One of the mirror lines is the line 𝑥 = −1.

6. What will be the coordinates the center of circle if it is translated to the left until it coincides with the

image resulting from two successive reflections of the two mirror lines?

𝑀1 𝑀2

Day 49 Practice


7. What will be the coordinates the center of circle if it is translated to the right until it coincides with

the image resulting from two successive reflections of the two mirror lines?

Two mirror lines at the lines 𝑦 = −𝑥 + 3 and 𝑦 = 𝑥 + 1 successively reflects a square with its vertex at

point V(1,1). Use this iformation to answer questions 8 and 9.

8. If these reflections are described by a rotation, what will be the coordinates of the center of rotation?

9. If these reflections are described by a rotation, what will be the angle of rotation?

Day 49 Practice


Use the graph below to answer questions 10-13.

10. Describe a single rigid motion which can map ∆𝐴𝐵𝐶 to its image.

11. Identify to mirror lines which can successively map ∆𝐴𝐵𝐶 to its image.

12. ∆𝐴𝐵𝐶 is successively reflected by two mirror lines at 𝑥 = −1 and 𝑥 = −4. On the graph, indicate the

image of ∆𝐴𝐵𝐶 to its left that results from two successive reflections of the two mirrors.

13. Describe a translation which can descried the series of reflections in NO. 12.

14. A student was passing through the center of a square room with vertical plane mirrors on opposite

sides of the wall. The room was measuring 10ft by 10ft. He noticed that multiple images were formed.

How far from the student was the image which could appear exactly as a translation.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A B

C

𝐶′

𝐵′ 𝐴′

Day 49 Practice


15. A barber fitted plane mirrors on two adjacent walls of his square barber shop. As he stood at the

center of his shop he noticed that there were three images of his body formed by the plane mirrors.

Which single rigid motion could map his body with the central image?

Use the following information to answer questions 16,17 and 18.

An hotelkeeper fitted his rectangular hotel with vertical plane mirrors on opposite walls. The distance

between the two opposite walls was 12ft. He then placed a table at the center of the hotel and noticed

that multiple images of the table formed on the plane mirrors. Two images with the same orientation as

the original table formed on both sides of the wall.

16. How long was the distance between the table and one image which had the same orientation as the

table.

17. What was the shortest distance between the two images which had the same orientation as the

table.

18. How far was one of the images from the wall.

In questions 19 and 20 state whether the given statement is true or false.

19. Any rotation can be written as two successive reflections.

20. A rotation of 45° about the origin can be described by two successive reflections with mirrors lines

intersected at the origin and making an angle of 90° with each other.

Day 49 Practice


Answer Keys Day 49:

1. 4𝑖𝑛

2. 3𝑖𝑛

3. 5in

4. 2𝑖𝑛

5. 4𝑖𝑛

6. (−4,0)

7. (4,0)

8. (1,2)

9. 180°

10. A rotation of 180° about the point (0,1)

11. 𝑥 = 0, 𝑦 = 1

12.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A B

C

𝐶′

𝐵′ 𝐴′

Day 49 Exit Slip


1. In the graph below, ∆𝐴𝐵𝐶 is mapped onto ∆𝐴′𝐵′𝐶′ by a rotation of 180° about the point (0,1). Write

the equation of two mirror lines which can describe the transformation.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A

B C

𝐴′

𝐶′ 𝐵′

Day 49 Exit Slip


Answer Keys

Day 49:

1. 𝑦 = 1

𝑥 = 0

80

High School Math Teachers

Geometry

Weekly Assessment Package

Week 10

©2020HighSchoolMathTeachers

81

Week 10

Weekly Assessments

82

Week #10 1. Construct an equilateral triangle inscribed in a circle of radius 1 in.

2. Construct a square inscribed in a circle of diameter 2.5 in

3. Construct a regular hexagon inscribed in a circle of radius 1.2 in

4. a. Identify four rigid motions

b. Identify those that preserves orientations and those that do not. c. Identify those that have fixed point(s) and those that do not.

5. An geometric figure is reflected about

𝑦 = 𝑥 then rotated by 90°about (1,2).

a. Compare the sizes of the two

images

b. Comment on the orientation of

the two images

c. Compare the interior angles of

the images.

6. Identify all fixed point on the figure

ABCD id reflected about line M.

B

A

C

D E

M

83

Week 10 - Keys

Weekly Assessments

84

Week #10 KEY 1. Construct an equilateral triangle inscribed in a circle of radius 1 in.

2. Construct a square inscribed in a circle of diameter 2.5 in

3. Construct a regular hexagon inscribed in a circle of radius 1.2 in 0722 486064/0723

003357

85

4. a. Identify four rigid motions

Rotation Translation Refection Glide reflection b. Identify those that preserves orientations and those that do not. Translation c. Identify those that have fixed point(s) and those that do not. Rotation Reflection Glide reflection

5. An geometric figure is reflected about

𝑦 = 𝑥 then rotated by 90°about (1,2).

a. Compare the sizes of the two

images

They have the same size

b. Comment on the orientation of

the two images

They have different orientations

c. Compare the interior angles of

the images.

The corresponding angles are equal

6. Identify all fixed point on the figure

ABCD id reflected about line M.

Fixed points are points that will not move after the transformation. These are all points on ABCD that lie on M.

B

A

C

D E

M

Day 51 Bellringer


1. Identify rigid motions that preserves the orientation of the pre image

2. Identify rigid motions that do not preserves the orientation of the pre image.

3. Why is dilation not a rigid motion?

4. How can you describe parallel lines

5. What is common about corresponding angles of parallel line?

Day 51 Bellringer


Answer Key

Day 51

1. Translation

2. Rotation, reflection and glide reflection

3. Because it involves change in size of the pre-image hence not qualifying to be a rigid motion

where sizes does not change

4. They are lines in the same plane that do not meet, they maintain the same distance along

their length

5. They face in the same direction and they are equal.

Day 51 Activity


1. Using the rubber bands, tied the three bars to form a triangle.

2. Place the triangle on the ground and draw its image, label it. ABC.

3. Take it and drop it to the ground with any angle or side facing any direction. Draw its image then

label it DEF. In each case, not the side with the same color and the initial one.

4. Take it and drop it to the ground with any angle or side facing any direction. Draw its image then label

it GHI. In each case, not the side with the same color and the initial one.


it JKL. In each case, not the side with the same color and the initial one.


it MNP. In each case, not the side with the same color and the initial one.


it QRS. In each case, not the side with the same color and the initial one.

7. Proceed with the similar procedure until you get as much triangle labels as possible.

Day 51 Activity


8. List all sides with the same color.

9. A part from color, Identify any geometrical features that the sides with the same colour have?

10. What is the best name to describe the solution in 9 above.

Day 51 Activity


In this activity, we would like to illustrate the idea of congruent sides using rigid motions of a simply

constructed triangle. Students will work in groups of at least 5. Each group is required to have three

plastic thin bars of any size between 2 in and 10 in (or any thin bar that can be identified to be different)

rubber bands and an open field where we can scribble down or any field together with whitewash to

make us make drawings.

Answer Keys

Day 51:

1 – 2.No response

3 -7. Different responses

7. Difference responses

8. They should be in 3 different groups

9. They are equal

10. Congruent sides

Day 51 Practice



Use each diagram to answer two questions after it.

1. Identify the most common rigid motion(s) relating the two figures in present.

2. Identify the congruent sides



A B

C

D

E R

W

X

Z

Y

S

U

S Q

E

Y

X

L

Day 51 Practice







W

E R

T

Y U

P A

S

D F

Day 51 Practice





13. Identify a single rigid motion relating the two figures in present.


F

G

H

J

K

L

X Z

L K

H J

Day 51 Practice


15. Identify the congruent sides based on the motion in 13 above.

Use the following image to answer questions 16 – 20

The figure below shows the motion of an equilateral triangle.

16. Identify one single rigid motion relating the two triangles above.

17. Identify the congruent sides for the rigid motion above

18. Identify another single rigid motion relating the two triangles above

19. Identify the congruent sides for the rigid motion above

20. Identify a single rigid motion that cannot map one triangle to the other in the figure above.

A

B

H

Y

C G

Day 51 Practice


Answer Keys

Day 51

1. Rotation

2. AC and DR,CB and RE, and BA and ED

3. Glide reflection

4. WX and SU, XZ and UY, and ZW and YS

5. Rotation and translation

6. SQ and XY, QE and YL, and ES and LX

7. Translation

8. RE and UY, EW and YT, and WR and TU

9. Reflection

10. FS and PS, SD and SA, and DF and AP

11. Rotation

12. LK and JH, KG and HF, and GL and FJ


14. Rotation and translation

15. HJ and XZ, JK and ZL, and KH and LX

16. Reflection or Translation

17. If reflection AB and HY, BC and YG, and CA and GH

If translation AB and GY, BC and YH, and CA and HG 18. Reflection or Translation

19. If reflection AB and HY, BC and YG, and CA and GH

If translation AB and GY, BC and YH, and CA and HG 20. Rotation

Day 51 Exit Slip


Identify corresponding sides in the following triangles if they are related by a series of rigid motions.

M N

L

D

K

Z

Day 51 Exit Slip


Answer Keys

Day 51

ML and DZ

LN and KD

MN and KZ

Day 52 Bellringer


1. The diagram below shows Δ PQR mapped onto Δ XYZ after undergoing a rigid motion. Study it in

order to answer the questions that follow.

(a) Which type of rigid motion has mapped Δ PQR onto Δ XYZ above?

(b) Identify congruent sides from the pre-image and the image?

P

Q R

𝑙 X

Y Z

Day 52 Bellringer


(c) Based on the type of rigid motion you have identified in (a) above, how does Δ PQR and Δ XYZ

compare?

2. Rectangle ABCD below has undergone a translation towards the right to form its image, rectangle

KLMN. Study the two rectangles and answer the following questions.

(a) Identify the corresponding sides on rectangles ABCD and KLMN above.

(b) Identify pairs of equal sides on rectangles ABCD and KLMN above.

A

B C

D K

L M

N

Day 52 Bellringer


Answer Keys Day 52:

1. (a) A reflection

(b) PQ and XY; QR and YZ; PR and XZ

(c)Δ PQR ≅ Δ XYZ, they have the same shape and size

2. (a) AB and KL; BC and LM; CD and MN; AD and KN

(b) AB and KL; BC and LM; CD and MN; AD and KN

Day 52 Activity


1. Using a ruler draw any triangle of a convenient size on the plain paper and label it ΔABC in the order

shown below.

2. Place the tracing paper on the plain paper and carefully trace ΔABC with the help of a straightedge.

Label the duplicate triangle ΔKLM in the order shown below.

3. Slide the tracing paper carefully over the plain paper until ΔABC and ΔKLM coincide. What does this

tell you about the relationship between their sizes and shapes?

4. Using a protractor measure ∠A then measure ∠K and write down their measures. What do you notice

about these two angles? Note the relative positions of the angles with reference to each other in the

two triangles.

5. Now measure ∠B then measure ∠L and write down their measures. What do you notice about these

two angles? Note the relative positions of the angles with reference to each other in the two triangles.

A

B C

K

L M

Day 52 Activity


6. Similarly, measure ∠C then measure ∠M and write down their measures. What do you notice about

these two angles? Note the relative positions of the angles with reference to each other in the two

triangles.

7. If we were to measure the lengths of the corresponding sides using a ruler, just like we have

measured the angles, what is the most likely outcome?

Day 52 Activity


In this activity students will work in groups of four to identify congruent angles in congruent triangles.

Students in the respective groups will require a protractor, a straightedge and A4 size tracing paper and

an A4 plain paper.

Answer Keys

Day 52:

1. No response

2. No response

3. They have the same size and shape? They are congruent

4. The angles are approximately equal. Students should note that these are corresponding angles.



7. The corresponding sides will have approximately the same length.

Day 52 Practice


On the grid below Δ KLM has been mapped onto Δ PQR after a half turn about the origin (0,0). Study it

and answer questions 1-5.

1. Identify all pairs of corresponding sides.

2. Identify all pairs of corresponding angles.

3. Show the mapping of corresponding vertices.

K

L

y

x

M

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

P

Q R

Day 52 Practice


4. Identify all pairs of congruent angles.

5. Basing on questions 1-4 above, what do you conclude about the two triangles?

The figure below shows two triangles. ΔPQR has been mapped to ΔABC after a translation. Study it and

use it to answer questions 6-10.

6. Identify all pairs of corresponding sides.

7. Identify all pairs of corresponding angles.

8. Identify all pairs of congruent angles.


10. What can you say about the sizes and the shapes of the two triangles with reference to each other?

A

B C

P

Q R

Day 52 Practice


ΔPQR has been mapped to ΔKLM through a reflection along the mirror line 𝑙 as shown below. Study the

figure and use it to answer questions 11-15.

11. List all pairs of corresponding sides.

12. List all pairs of corresponding angles.

13. List all pairs of congruent angles.


15. According to what you have learnt on rigid motion do you expect the two triangles to be congruent?

Give a reason.

A

B C

P

Q R

𝑙

Day 52 Practice


A student performed a glide reflection on ΔPQR and it was mapped onto ΔXYZ as shown below. Study

the student’s diagram and answer questions 16-20.

16. List all pairs of corresponding sides.

17. List all pairs of corresponding angles.

18. List all pairs of congruent angles.

X

Z Y

P

Q R

𝑙

Day 52 Practice



20. Is the dotted triangle congruent to both ΔPQR and ΔXYZ? Give a reason.

Day 52 Practice


Answer keys Day 52:

1. KL and PQ

LM and QR

KM and PR

2. ∠K and ∠P

∠L and ∠Q

∠M and ∠R

3. K → P

L → Q

M → R

4. ∠K and ∠P

∠L and ∠Q

∠M and ∠R

5. The two triangles are congruent.

6. AB and PR

BC and RQ

AC and PQ

7. ∠A and ∠P

∠B and ∠R

∠C and ∠Q

8. ∠A and ∠P

∠B and ∠R

∠C and ∠Q

9. A → P

B → R

C → Q

10. They have the same size and shape.

11. AB and PQ

BC and QR

AC and PR

12. ∠A and ∠P

∠B and ∠Q

∠C and ∠R

13. ∠A and ∠P

∠B and ∠Q

∠C and ∠R

14. A → P

B → Q

C → R

15. Yes. Rigid motion preserves distances.

16. PQ and XZ

QR and ZY

PR and XY

Day 52 Practice


17. ∠P and ∠X

∠Qand ∠Z

∠R and ∠Y

18. ∠P and ∠X

∠Qand ∠Z

∠R and ∠Y

19. P → X

Q → Z

R → Y

20. Yes. Rigid motions preserve size and shape.

Day 52 Exit Slip


On the grid below Δ PQR has been mapped onto Δ ABC after a half turn about the origin (0,0).

Considering Δ PQR and Δ ABC, identify all the pair of congruent angles.

P

Q

y

x

R

−6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7

5

4

3

2

1

−1 A

C B

Day 52 Exit Slip


Answer Keys

Day 52:

∠P ≅ ∠A

∠Q ≅ ∠C

∠R ≅ ∠B

Day 53 Bellringer


1. State whether each of the following pairs of triangles is congruent or not.

a)

b)

Day 53 Bellringer


c)

d)

2. A triangle is rotated clockwise about the origin and then reflected over the y-axis. Is the final image

congruent to the original triangle.

Day 53 Bellringer


Answer Key Day 4

1. a) Yes

b) No

c) Yes

d) Yes

2. Yes

Day 53 Activity


1. Place tile of the plane paper and draw its image accurately. Label it.

2. Lift it and place it at another position in any direction and draw its image accurately. Label it.

3. Before looking at the sides and angles, what makes you conclude that the two triangles drawn are

congruent?

4. Identify corresponding angle and sides

5. Measure all angles all sides

6. Compare corresponding angles. What do you notice about each pair of corresponding angles?

7. Measure the corresponding sides

8. What do you notice about each pair of corresponding sides?

9. Based on the results above, make a generalization of the features congruent triangles must satisfy.

Day 53 Activity


In this activity, we would like to show that if two triangles are congruent, then they have corresponding

angles and sides. Students will work in groups of at least 5. Each group will a triangular small tile or a

triangular framework of rigid wires, a plane paper, ruler, protractor and a pencil.

Answer Keys

Day 53:

1 – 2. Different responses

3. They are congruent because drawn using the same tile

4 - 5. Different responses

6. They are equal

7. Different responses

9. They are equal

Day 53 Practice


Questions 1 and 3 tests on the understanding of the theorem on congruence

State the conditions that two triangles must satisfy so that they may be congruent.

1. First Condition

2. Second condition

3. Given that the two conditions above are satisfied, what can we conclude about the scale factor if the

triangles?

4. Which transformations are used to show that two triangles are congruent?


We want to prove that triangles ACB and QPR are congruent.

Fill the following table

STATEMENT REASON/ IDENTICATION

AC and QP, CB and PR, BA and RQ 5.

6. Corresponding angles

𝐴𝐶 = 𝑄𝑃, 𝐶𝐵 = 𝑃𝑅, 𝐵𝐴 = 𝑅𝑄 7.

∠𝐴 = ∠𝑄, ∠𝐶 = ∠𝑃, ∠𝐵 = ∠𝑅 8.

So far, the two triangles related by dilation 9. 𝐴𝐶

𝑄𝑃=

𝐶𝐵

𝑃𝑅=

𝐵𝐴

𝑅𝑄= 1

10.

Triangles have a linear scale factor of 1 𝟏𝟏.

The Area scale factor is also 1 12.

Triangles have equal area 13.

The triangles are congruent 14.

5

A

B

C

Q

R

P

10

10

5

8.7

8.7

60°

60° 30°

90°

Day 53 Practice



We would like to prove that the corresponding sides and angles of congruent triangles are equal if the

triangles are congruent.

Fill the following table


The triangles are congruent 15. Given

One triangle can be mapped on the other after one or more rigid motions.

16.

Thus, corresponding angles are equal 17.

And corresponding angles are equal 18.

Use the following information to answer questions 19 – 20.

The two triangles above are congruent

19. Find the value of x

20. Find the length of KY.

Y

S K

Y

S K

2𝑥 + 1 𝑖𝑛

4𝑥 − 3 𝑖𝑛

Day 53 Practice


Answer Keys

Day 53

1. Corresponding angles are equal or responding sides are equal

2. Corresponding angles are equal or responding sides are equal

3. 1

4. Glide reflection, reflection, rotation, translation


AC and QP, CB and PR, BA and RQ 5. Corresponding angles

6. ∠𝑨 and ∠𝑸, ∠𝑪 𝐚𝐧𝐝 ∠𝑷, ∠𝑩 𝐚𝐧𝐝 ∠𝑹 Corresponding angles

𝐴𝐶 = 𝑄𝑃, 𝐶𝐵 = 𝑃𝑅, 𝐵𝐴 = 𝑅𝑄 7. Comparing their lengths

∠𝐴 = ∠𝑄, ∠𝐶 = ∠𝑃, ∠𝐵 = ∠𝑅 8. Comparing their angular measurements

So far, the two triangles related by dilation 9. Corresponding angles are equal

𝐴𝐶

𝑄𝑃=

𝐶𝐵

𝑃𝑅=

𝐵𝐴

𝑅𝑄= 1

10. Upon modification of 𝑨𝑪 = 𝑸𝑷, 𝑪𝑩 = 𝑷𝑹, 𝑩𝑨 = 𝑹𝑸

Triangles have a linear scale factor of 1 𝟏𝟏.

𝑨𝑪

𝑸𝑷=

𝑪𝑩

𝑷𝑹=

𝑩𝑨

𝑹𝑸= 𝟏

The Area scale factor is also 1 12. (𝒍𝒊𝒏𝒆𝒂𝒓 𝒔𝒄𝒂𝒍𝒆 𝒇𝒂𝒄𝒕𝒐𝒓)𝟐 =𝟏

Triangles have equal area 13. Area scale factor is 1

The triangles are congruent 14. Area scale factor is 1 and corresponding angles are equal


The triangles are congruent 15. Given

One triangle can be mapped on the other after one or more rigid motions.

16. Properties of rigid motion

Thus, corresponding angles are equal 17. Properties of rigid motion

And corresponding angles are equal 18. Properties of rigid motion

19. 2 in 20. 5 in

Day 53 Exit Slip


Based on the definition of congruence using rigid motion, Identify the condition that will make the two

triangles below to congruent.

R J

E

Q

F X

Day 53 Exit Slip


Answer Keys

Day 53

The following angles and sides a must be equal

RE and XQ

EJ and QF

JR and FX

AND

Angles ∠𝐸 and ∠𝑄, ∠𝑅 and ∠𝑋, and ∠𝐹 and ∠𝐽

Day 54 Bellringer


1. Use the figure below to answer the questions that follow. ∆𝐴𝐵′𝐶′ is an image of ∆𝐴𝐵𝐶.

a) Find the center of dilation.

b) Find the linear scale factor.

2. Points 𝐴(1,1), 𝐵(3,1), 𝐶(1,3), 𝐷(3,3) are vertices of a square. Taking the origin as the center of

dilation, find the vertices of the image when the scale factor is :

a) 2

b) 1

c) 3

𝐵′ 𝐵 A

𝐶′

C

1in

2in

Day 54 Bellringer


Answer Key

Day 54

1. a) Point A

b) 1

2

2. a) 𝐴(2,2), 𝐵(6,2), 𝐶(2,6), 𝐷(6,6)

b) 𝐴(1,1), 𝐵(3,1), 𝐶(1,3), 𝐷(3,3)

c) 𝐴(3,3), 𝐵(9,3), 𝐶(3,9), 𝐷(9,9)

Day 54 Activity


1. Plot a graph with a scale of 1 square representing 1 unit as shown.

2. Draw a triangle with vertices at points A(1,2), B(6,1) and C(4,4) as shown

Day 54 Activity


3. Draw straight lines A to pass through the origin.

4. Position the compass at the origin and extend it to point A then using the same width make an arc on

opposite side to cross the line as shown.

5. Draw lines from points B and C through the origin and on each line, repeat step 3 as shown below.

Day 54 Activity


6. Join the intersections of the arcs with straight lines as shown below.

7. Measure the distance of each point from the origin with its image distance from the origin.

8. Are they equal?

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A

B

C

Day 54 Activity


In this activity students will draw a triangle and its image under a dilation with a scale factor of -1.

Student will work in groups of at least three and each group is required to have a graph paper, a pencil,

a ruler and a compass.

Answer Keys

Day 54:

1-6. No response

7. Different responses

8. Yes

Day 54 Practice


Use the diagram below to answer questions 1-3.

The rectangle below is dilated with a scale factor of -1 about point R.

(the figure is not drawn to scale)

1. Draw the image

2. Find the distance 𝑅𝑇′

3. Find the distance 𝑈𝑈′

∆𝑀𝑁𝑂 is dilated with a scale factor of -1 about point P. The distances MP, NP and OP are 3in, 2in and

5in respectively. Use this information to answer questions 4-6.

4. What is the distance of 𝑂𝑃′?

5. What is the distance of 𝑁𝑃′?

6. What is the distance of 𝑀𝑃?

R

U T

S

6𝑖𝑛

8𝑖𝑛

Day 54 Practice


Use the graph below to answer questions 7-19. ∆𝐴𝐵𝐶 is dilated with a scale factor of -1 with the center

at the origin.

7. What will be the coordinates of the image of point A?

8. What will be the coordinates of the image of point B?

9. What will be the coordinates of the image of point C?

10. On the graph, indicate the image of ∆𝐴𝐵𝐶.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A

B

C

O

Day 54 Practice


With respect to the graph have drawn in question 10 above, state whether the statements in questions

11-14 are true or false.

11. The distances OB and 𝑂𝐵′ are equal.

12. 𝑂𝐵 =1

2𝐵𝐵′

13. 1

2𝑂𝐴 = 𝐴𝐴′

14. 𝑂𝐶 = 𝑂𝐶′

15. 1

2𝑂𝐶 = 𝐶𝐶′

16. 𝑂𝐴 ≅ 𝑂𝐴′

17. 𝑂𝐵 ≅ 𝑂𝐵′

18. 𝑂𝐶 ≅ 𝑂𝐶′

19. 𝑂𝐵 ≅ 𝐵𝐵′

20. The triangle below is translated with a scale factor of -1 and center O. Indicate the image.

O

Day 54 Practice


Answer Key

Day 54:

1.

2. 10𝑖𝑛

3. 12𝑖𝑛

4. 5𝑖𝑛

5. 2𝑖𝑛

6. 3𝑖𝑛

7. (−1, −1)

8. (−4, −2)

9. (−1, −4)

R

U T

S

6𝑖𝑛

8𝑖𝑛

𝑇′ 𝑈′

𝑅′ 𝑆′

Day 54 Practice


10.

11. True

12. True

13. False

14. False

15. False

16. True

17. True

18. True

19. False

20.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A

B

C

𝐴′

𝐶′

𝐵′

O

Day 54 Exit Slip


The triangle below is dilated about point P with a scale factor of -1. Indicate the image.

P

Day 54 Exit Slip


Answer Keys

Day 54:

P

136


Geometry


Week 11


137

Week 11

Weekly Assessments

138

Week #11 1. Use the triangles below to answer the

questions that follow

a) Which rigid motion will map the two

triangle onto each other?

b) What is the size of ∠𝑀?

c) What is the size of ∠𝑂?

d) Is ∆𝐴𝐵𝐶 ≅ ∆𝑀𝑁𝑂?

2. Construct squares inscribed in each circle below.

a) A circle with a radius of 0.8 𝑖𝑛. b) A circle with a radius of 1 𝑖𝑛.

A B M N

O C

60°

139

3. ∆𝐴𝐵𝐶 is dilated about point C with a scale factor of -1 as shown. a) What is the length of 𝐴′𝐶? b) What is the length of 𝐵′𝐶? c) What is the length of 𝐴′𝐵′? d) Is ∆𝐴𝐵𝐶 ≅ ∆𝐴′𝐵′𝐶?

4. Use the diagram below to answer the questions that follow. a) Which rigid motion will map the triangles above onto each other? b) Which side is congruent to AC? C) Which side is congruent to AB? d) Which angle is congruent to ∠𝐶?

A

B

C

𝐴′

𝐵′ 2.5 𝑖𝑛

3 𝑖𝑛

1.8 𝑖𝑛 A B

C

M N

O

140

5. Use the triangles below to answer the questions that follow. a) State whether the triangles above are congruent. b) Give a reason for the answer you gave in (a) above.

6. Use the diagram below to answer the questions that follow. a) State a single rigid motion that will map the two rectangles. b) Describe a series of reflections that will map the two rectangles.

2 𝑖𝑛

4 𝑖𝑛

4 𝑖𝑛

3 𝑖𝑛

5 𝑖𝑛

𝐴 𝐵

𝐷 𝐶

𝐵′ 𝐴′

𝐶′ 𝐷′

141

Week 11 - Keys

Weekly Assessments

142

Week #11 KEY

3. Use the triangles below to answer the

questions that follow

e) Which rigid motion will map the two

triangle onto each other?

Reflection f) What is the size of ∠𝑀?

60°

g) What is the size of ∠𝑂?

30°

h) Is ∆𝐴𝐵𝐶 ≅ ∆𝑀𝑁𝑂?

Yes

4. Construct squares inscribed in each circle below.

a) A circle with a radius of 0.8 𝑖𝑛. b) A circle with a radius of 1 𝑖𝑛.

A B M N

O C

60°

143

3. ∆𝐴𝐵𝐶 is dilated about point C with a scale factor of -1 as shown. a) What is the length of 𝐴′𝐶? 2.5 𝑖𝑛 b) What is the length of 𝐵′𝐶? 1.8 𝑖𝑛 c) What is the length of 𝐴′𝐵′? 3 𝑖𝑛 d) Is ∆𝐴𝐵𝐶 ≅ ∆𝐴′𝐵′𝐶? Yes

4. Use the diagram below to answer the questions that follow. a) Which rigid motion will map the triangles above onto each other? Glide reflection b) Which side is congruent to AC? MO C) Which side is congruent to AB? MN d) Which angle is congruent to ∠𝐶?

∠𝑂

A

B

C

𝐴′

𝐵′ 2.5 𝑖𝑛

3 𝑖𝑛

1.8 𝑖𝑛 A B

C

M N

O

144

5. Use the triangles below to answer the questions that follow. a) State whether the triangles above are congruent. They are not congruent b) Give a reason for the answer you gave in (a) above. All coresponding sides are not equal.

6. Use the diagram below to answer the questions that follow. a) State a single rigid motion that will map the two rectangles. A reflection b) Describe a series of reflections that will map the two rectangles. A reflection over a horizontal mirror line below ABCD followed a reflection over a vertical mirror line on the right then a reflection over a horizontal mirror line.

2 𝑖𝑛

4 𝑖𝑛

4 𝑖𝑛

3 𝑖𝑛

5 𝑖𝑛

𝐴 𝐵

𝐷 𝐶

𝐵′ 𝐴′

𝐶′ 𝐷′

Day 56 Bellringer


ΔABC is mapped onto ΔPQR after a translation to the right as shown below. Identify angles and sides

corresponding to the ones indicated.

(a) AB̅̅ ̅̅

(b) QR̅̅ ̅̅

(c) AC̅̅̅̅

(d) ∠P

(e) ∠B

(f) ∠C

A

B C

P

Q R

Day 56 Bellringer


Answer Keys

Day 56:

(a) PQ̅̅̅̅

(b) BC̅̅̅̅

(c) PR̅̅̅̅

(d) ∠A

(e) ∠Q

(f) ∠R

Day 56 Activity


1. Using a ruler draw any triangle of a convenient size on the plain paper and label it ΔPQR in the order

shown below.

2. Place the tracing paper on the plain paper and carefully trace ΔPQR with the help of a straightedge.

Label the duplicate triangle ΔXYZ in the order shown below.

3. Assuming that ΔPQR is mapped onto ΔXYZ by a translation, identify all pairs of corresponding sides

from the triangles.

4. Using a ruler, measure the length of each side in all the corresponding pairs of sides you have

identified in question 3 above in inches and write down the measurements. What do you notice about

the relationship between the lengths of corresponding sides?

5. What can you say about the relationship between the length of corresponding sides in ΔPQR and

ΔXYZ and the congruency of these triangles?

P

Q R

X Y

Z

Day 56 Activity


In this activity students will work in groups of at least four to discover the congruence postulate SSS

using congruent triangles. Each groups will require a ruler, an A4 size tracing paper and an A4 plain

paper.

Answer Keys

Day 56:

1. No response

2. No response

3. PQ and ZX

QR and XY

PR and ZY

4. All the corresponding sides have the same length; all the corresponding sides are congruent.

5. All the corresponding sides in the two triangles are congruent hence the two triangles are congruent.

Day 56 Practice


The pairs of triangles in questions 1-5 are congruent. Find the length of the left-out third side. All the

measurements are in inches. Note that the triangles are not drawn to scale.

1.

2.

7 9 9

8 8

13

13

12

10

12

Day 56 Practice


3.

4.

4

6

5

6 5

4 4

14

13

13

Day 56 Practice


5.

6. The figure below shows parallelogram PQRS, identify two pairs of congruent triangles using the SSS

postulate.

7. The figure below shows square JKLM. Show that ΔJLK ≅ ΔJLK using the SSS postulate.

8

8

10

5

10

P

Q R

S

T

J

K L

M

Day 56 Practice


The figure below shows two congruent triangles, ΔABC and ΔADE. Use it to answer questions 8 and 9.

8. List all pairs of corresponding sides of the two triangles.

9. Show that the two triangles are congruent.

A

B C

D E 6 in.

6 in. 10 in.

10 in.

Day 56 Practice


Use the figure below to answer questions 10-16. PQ̅̅̅̅ ≅ PR̅̅̅̅ and PT̅̅̅̅ bisects QR̅̅ ̅̅ .

10. Identify two congruent triangles in the figure above.

11. List the pairs of corresponding sides in the two triangles you have identified in question 10 above.

Complete the table below using the figure above.

Statement Reason

PQ̅̅̅̅ ≅ PR̅̅̅̅ Given

12. Given

Point T is the midpoint of QR̅̅ ̅̅ 13.

QT̅̅ ̅̅ ≅ RT̅̅̅̅ 14.

15. Reflexive property

16. The S.S.S postulate

P

Q R T

Day 56 Practice


Study the figure below and use it to answer questions 17-20. Point C is the common point of the two

triangles.

Complete the table below by identify a side corresponding to the one given

AB̅̅ ̅̅ 17.

18. DE̅̅ ̅̅

19. If BC̅̅̅̅ ≅ CD̅̅̅̅ , state whether these triangles will be congruent or not.

20. Give a reason for you answer to question 19 above.

A

B

C

D

E

Day 56 Practice


Answer keys

Day 56:

1. 7 in.

2. 10 in.

3. 4 in.

4. 14 in.

5. 5 in.

6. ΔPTS and ΔQTR

ΔPTQ and ΔRTS

ΔPQS and ΔQRS

ΔPRS and ΔPQR

7. JM̅̅̅̅ ≅ LK̅̅̅̅ (opposite sides of a square are congruent)

ML̅̅ ̅̅ ≅ KJ̅ (opposite sides of a square are congruent)

JL̅ ≅ LJ̅ (Reflexive property)

Therefore S.S.S postulate satisfied.

8. BC̅̅̅̅ and AC̅̅ ̅̅

AC̅̅̅̅ and DE̅̅ ̅̅̅

AB̅̅ ̅̅ and DA̅̅ ̅̅̅

9 . BC̅̅̅̅ ≅ AC̅̅ ̅̅ (Given)

AC̅̅̅̅ ≅ DE̅̅ ̅̅̅ = 6 𝑖𝑛

AB̅̅ ̅̅ ≅ DA̅̅ ̅̅̅ = 10 𝑖𝑛. The S.S.S postulate is achieved.

10. ΔPQT and ΔPRT

Day 56 Practice


11. PT̅̅̅̅ and PT̅̅̅̅

PQ̅̅̅̅ and PR̅̅ ̅̅

QT̅̅ ̅̅ and RT̅̅̅̅̅

12.

Statement Reason

PQ̅̅̅̅ ≅ PR̅̅̅̅ Given

12. 𝐏𝐓̅̅ ̅̅ bisects QR Given

Point T is the midpoint of QR̅̅ ̅̅ 13. 𝐏𝐓̅̅ ̅̅ is the perpendicular bisector of 𝐐𝐑̅̅ ̅̅ ̅

QT̅̅ ̅̅ ≅ RT̅̅̅̅ 14.T is the midpoint of 𝐐𝐑̅̅ ̅̅

15. 𝐏𝐓̅̅ ̅̅ ≅ 𝐏𝐓̅̅ ̅̅ Reflexive property

16. 𝚫𝐏𝐐𝐓 and 𝚫𝐏𝐑𝐓 The S.S.S postulate

17.

AB̅̅ ̅̅ 17. 𝐂𝐄̅̅̅̅

18. 𝐂𝐀̅̅ ̅̅ DE̅̅ ̅̅

19. They will be congruent.

20. All the three corresponding sides will be congruent/ according to the S.S.S postulate.

Day 56 Exit Slip


Consider triangles ΔXYZ and ΔKLM. The measurements are in inches.

Are the two triangles congruent? If so, show why?

5

3

5 3

4

4

X

Y Z

K

L

M

Day 56 Exit Slip


Answer Keys Day 56:

Yes.

XY̅̅̅̅ = MK̅̅̅̅̅ = 3 𝑢𝑛𝑖𝑡𝑠

YZ̅̅̅̅ = KL̅̅̅̅ = 4 𝑢𝑛𝑖𝑡𝑠

XZ̅̅̅̅ = ML̅̅ ̅̅ = 5 𝑢𝑛𝑖𝑡𝑠.

SSS postulate is achieved thus ΔXYZ ≅ ΔKLM

Day 57 Bellringer


1. Find the third angle of a triangle whose angles are given below.

(i). 77° 𝑎𝑛𝑑 46°

(ii). 63° 𝑎𝑛𝑑 81°

(iii). 92° 𝑎𝑛𝑑 64°

2. Given that the corresponding sides of a triangle are 12 𝑖𝑛 and 15 𝑖𝑛 respectively.

(i). Find the linear scale factor relating the two given the smaller triangle is the image of the larger one.

(ii). Find the measurement of the other side of the smaller triangle is the corresponding side of the

bigger ttriangle is 25 in.

Day 57 Bellringer


Answer Key

Day 57

1. (i). 57°

(ii). 36°

(iii). 24°

2. (i) 4

5

(ii). 20𝑖𝑛

Day 57 Activity


1. Draw a 4 in line and label it TY.

2. Using a protractor, construct an angle of 30° and 60° at T and Y respectively.

3. Let the lines from T and Y meet respectively at U.

4. Draw another 4 in line and label it PR.

5. Using a protractor, construct an angle of 30° and 60° at P and R respectively.

6. Let the lines from P and R meet respectively at W.

7. Measure the angles at U and W respectively, record your answer.

8. How are the angles in 7 above related?

9. Measure the length of lines TU and YU, as well as PW and RW, record your answer.

10. Find the area of triangles TYU and PRW.

11. Compare the areas in 10 above.

12. Make the conclusions about the results based on the information provided.

Day 57 Activity


In this activity, we would like to prove by construction that the postulate ASA is true, that is, once a

triangles satisfy the postulate, then the triangles inclined are congruent. Students will work in groups of

at least 3. Each group will need A4 plain paper, protractor, a ruler and a pencil.

Answer Keys

Day 57:

1 – 6.No response

7. ∠𝑈 = 90°, ∠𝑊 = 90

8. The angles are equal

9. 𝑇𝑈 = 3.5 in and 𝑌𝑈 = 2 𝑖𝑛

𝑃𝑊 = 3.5 in and 𝑅𝑊 = 2 𝑖𝑛

10. 𝑇𝑌𝑈 = 3.5 𝑠𝑞 𝑖𝑛 and 𝑃𝑅𝑊 = 3.5𝑠𝑞. 𝑖𝑛.

11.They are equal

12. Given two triangles whose two corresponding lines and corresponding angles at the end points are

equal, the triangles are congruent.

Day 57 Practice


1. Identify a property that is shared between dilated images those that are congruent.

2. Two images are related by a scale factor of 1, what would be the transformation between them?

Name all of them.


Two triangles are such that corresponding sides are congruent

3. Are they equilateral triangles? Why?

4. Comment of their angles

5. Is ASA satisfied?

6. Explain the answer above.

Use the following diagrams to answer questions 7 - 10 .

7. Identify all pairs of corresponding angles

Q P

F

B

T

G

6 6

Day 57 Practice


8. Identify all pairs of corresponding sides

9. Based on ASA postulate, are the triangle congruent?


Use the following diagrams to answer questions 11 – 14.

11. Identify all pairs of corresponding angles

12. Identify all pairs of corresponding sides

13. Based on ASA postulate, are the triangle congruent?


X

U

D

9

12

B 12

Day 57 Practice


Use the following diagram to answer questions 15 – 18.

In the figure below, SVR is an Isosceles triangle and J is the midpoint of line SR.

15. Write a relation between lines SJ and JR.

16. Write a relation between the angles SVR and JVR.

17. Identify all pairs of corresponding angles in triangle SVJ and JVR.

18. Identify all pairs of corresponding sides in triangle SVJ and JVR.

19. Based on ASA postulate, are the triangle, SVJ and JVR, congruent?


S

V

R

J

Day 57 Practice


Answer Keys

Day 57

1. Corresponding angles are equal

2. Reflection, rotation, glide reflection, translation and dilation of scale factor 1

3. No, No enough information is given to confirm if they are equilateral triangles


5. Yes

6. Since all corresponding sides are equal, we pick just one side. Since corresponding angles are

equal, the corresponding angles the end points of the chosen side will be congruent hence ASA

is satisfied.

7. QF and TB, FP and BG, and PQ and GT

8. ∠𝑄 and ∠𝑇, ∠𝐹 and ∠𝐵, ∠𝐺 and ∠𝑃

9. No

10. Corresponding angles equal, but no enough information on corresponding sides

11. ∠𝑋 and ∠𝑉, ∠𝑈 and ∠𝑁, ∠𝐷 and ∠𝐵

12. XU and VN, DU and NB, and DX and BV

13. YES

14. Corresponding sides DX and BV are equal and corresponding angles at their endpoints are

congruent

15. 𝑆𝐽 = 𝐽𝑅

16. ∠𝑆𝑉𝑅 = 2∠𝐽𝑉𝑅

17. ∠𝑅𝐽𝑉 and ∠𝑉𝐽𝑆, ∠𝑆 and ∠𝑅, ∠𝑅𝑉𝐽 and ∠𝐽𝑉𝑆

18. VJ and VJ (Common), JR and SJ, and RV and SV

19. Yes

20. Sides RV and SV are equal (Given). Since triangle SVR is Isosceles, and J is the midpoint of RS, JV

is the bisector of angle SVR, hence ∠𝑅𝑉𝐽 = ∠𝐽𝑉𝑆 and ∠𝑆 = ∠𝑅 being base angles of the

Isosceles triangle.

Hence ASA is achieved.

Day 57 Exit Slip


Identify a reason based on ASA postulate that will make the triangles not congruent.

55°

25°

4 in

25°

55°

4 in

Day 57 Exit Slip


Answer Keys

Day 57

Corresponding angles at the end points of the line equal to 4in are not congruent

Day 58 Bellringer


Find out if the two triangles given are congruent, hence identify the postulate used.

1.

2.

3.

t t

13

13

H

L

T

V

R

7

15°

7

15°

18°

4

18° 8

76°

86° 86°

76°

Day 58 Bellringer


Identify an included angle with respect to the sides whose measurements are given

4.

5.

A

S N

7 units

6.2 units

E

B

M

4.7

9.3

Day 58 Bellringer


Answer Key

Day 58

1. (i). They are congruent. Postulate SSS

(ii). They are congruent. Postulate ASA

(iii). They are not congruent

No two pairs of corresponding sides are given to help us establish the postulate

2. (i). Angle ANS

(ii). Angle BME

Day 58 Activity


1. Draw a 3.5 in line and label it AB.

2. Using a protractor, construct an angle of 90° at B.

3. Locate C so that BC is 6 in. Connect C to A

4. Draw another 3.5 in line and label it XY.

5. Using a protractor, construct an angle of 90° at Y..

6. Locate Z so that YZ is 6 in. Connect Z to X

7. Measure the angles at A and C respectively, record your answer.

8. Measure the angles at X and Z respectively, record your answer.

9. What do you conclude about the angles of the two triangles?

10. Find the area of the two triangles.

11.Compare the areas in 10 above.

12. Make the conclusions about the results based on the information provided.

Day 58 Activity


In this activity, we would like to prove by construction that the postulate SAS is true, that is, once a

triangles satisfy the postulate, then the triangles inclined are congruent. Students will work in groups of

at least 3. Each group will need A4 plain paper, protractor, a ruler and a pencil.

Answer Keys

Day 58:

1 – 6. No response

7. ∠𝐴 ≈ 60°, ∠𝐶 ≈ 30°

8.Corresponding angles are equal

9. Area of 𝐴𝐵𝐶 = 10.5 𝑠𝑞 𝑖𝑛 and Area of 𝑋𝑌𝑍 = 10.5𝑠𝑞. 𝑖𝑛.

11.They are equal

12. Given two triangles whose at least one corresponding angles are equal and the corresponding sides

bounding the angles are equal, then the triangles are congruent.

Day 58 Practice


Use SAS postulate to answer the following questions.

Use the diagrams above to answer questions 1 and 2

1. Are the two triangles above congruent?

2. Explain your answer.


3. Are the two triangles above congruent?

4. Explain your answer.

28°

7 11

28°

11

7

24°

12

6

24°

12

6

Day 58 Practice


Use the following diagrams to answer questions 5 - 7 .

5. Find the value of 𝑡.

6. Are the two triangles, above, congruent?

7. Explain your answer

Use the following diagrams to answer questions 8 - 14.

8. Find the size of angle T and V.

9. If 𝑈𝑇 =2

√3, find the value of ST.

60°

30° S

T

U

V

W

X

Day 58 Practice


10. If 𝑈𝑇 =2

√3, find the value of VW.

11. Find the value of WX.

12. What is the relationship between US and VX?

13. Find the value if VX.

14. Are the two triangle congruent?

15. Using SAS postulate where angle TSU is one of the included angle, prove your results in 13 above.

Use the following diagrams to answer questions 16 – 20.

We have that QR =2 in.

16. Find the length of RY.

45°

45° Y

Q

R

P

L

J

Day 58 Practice


17. Find the length of JP.

18. Find the length of PL

19. Find the area of both triangles

20. Make a conclusion based on the information given in before 16 above.

Day 58 Practice


Answer Keys

Day 58

1. YES

2. Two Corresponding sides and the corresponding included angle are congruent.

3. YES

4. Two Corresponding sides and the corresponding included angle are congruent.

5. 𝑡 = 112°

6. No

7. The corresponding included angles are not congruent

8. ∠𝑇 = 60°, ∠𝑉 = 30°

9. 4

√2

10. 2

11. 2

√3

12. No direct relationship before congruency is set up

13. 4

√2

14. Yes

15. 𝑈𝑇 = 𝑋𝑊 =2

√3

𝑈𝑆 = 𝑉𝑊(𝐺𝑖𝑣𝑒𝑛) 𝐼𝑛𝑐𝑙𝑢𝑑𝑒𝑑 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 Thus, SAS is satisfied

16. 2 in

17. 2 in

18. 2 in

19. Area of ∆𝑌𝑄𝑅 = 2 sq. in, Area of ∆𝐽𝑃𝐿 = 2 sq. in,

20. Right triangles whose pair of corresponding sides and angles are equal are congruent

Day 58 Exit Slip


Use SAS postulate to show that the triangles are congruent. Explain.

8 in

8 in

87°

48°

5in

A

B

C

87°

45°

5in

D

T

Y

Day 58 Exit Slip


Answer Keys

Day 58

∠𝐴 = ∠𝐷 = 180° − 87° − 45° = 48°

Corresponding sides are equal, 𝐵𝐴 = 𝑇𝐷 = 5 𝑖𝑛, 𝐴𝐶 = 𝐷𝑌 = 8 𝑖𝑛

Thus, the postulate, SAS is satisfied implying that the triangles are congruent.

Day 59 Bellringer


1. (a) Identify the included angle between PR̅̅̅̅ and QR̅̅ ̅̅ in ΔPQR below.

(b) Identify the included side between ∠𝑋 and ∠𝑌 in ΔXYZ below.

2. State the congruence postulate that makes the following triangles congruent.

(a)

P

Q

R

Y Z

X

Day 59 Bellringer


(b)

(c)

Day 59 Bellringer


Answer Keys

Day 59:

1. (a) ∠R

(b) XY̅̅̅̅

2. (a) S.A.S

(b) A.S.A

(c) S.S.S

Day 59 Activity


1. Using a ruler draw any triangle of convenient size on the plain paper provided and label it ΔABC.

Follow the order shown below.

2. Using a tracing paper, duplicate ΔABC and label it ΔKLM in the order shown below.

3. Using a ruler, measure the length of all the corresponding sides on ΔABC and ΔKLM and write down

their lengths. Compare the lengths of the corresponding sides and write down your conclusion.

4. Which triangle congruence postulate has been investigated in question 3 above?

5. Identify the included angle between sides AB and BC on ΔABC.

6. Similarly, identify the included angle between sides KL and LM on ΔKLM.

7. Measure each of the two included angles you have identified in questions 5 and 6 above using a

protractor. What do you notice?

A

B C

K

L M

Day 59 Activity


8. Basing on questions 5-7 above, which triangle congruence postulate is being investigated?

9. Identify the included side between ∠A and ∠B on ΔABC

10. Similarly, identify the included side between ∠K and ∠L on ΔKLM.

11. Use a protractor to measure ∠A and ∠K and compare their measures. What do you notice?

12. Measure the length of each of the two included sides you have identified in questions 9 and 10

above using a ruler and compare their lengths. What do you notice?

13. Basing on questions 9-12, which triangle congruence postulate is being investigated?

14. Considering the steps above, state whether the two triangles are congruent or not? Give a possible

reason.

Day 59 Activity


In this activity students will work in groups of two to four to verify the triangle congruence postulates,

S.S.S, S.A.S and A.S.A. Students in the groups will require a ruler, a protractor, a tracing paper of A4 size,

a plain paper of A4 size.

Answer Keys Day 59:

1. No response

2. No response

3. The corresponding sides are congruent

4. S.S.S postulate

5. ∠B

6. ∠L

7. The angles are congruent

8. S.A.S postulate

9. AB̅̅ ̅̅

10. KL̅̅̅̅

11. ∠A ≅ ∠K

12. They are congruent

13. A.S.A postulate.

14. They are congruent. The triangle congruence postulates are achieved.

Day 59 Practice


For questions 1-3, use the triangle congruence postulate indicated to mark the triangles given that in

each case. ΔPQR ≅ ΔKLM.

1. S.S.S

2. S.A.S

3. A.S.A

P

Q M L

K

R

P

Q M L

K

R

P

Q M L

K

R

Day 59 Practice


In questions 4 -10, identify a congruent triangle to the one indicated based. In each case name the

triangle using proper correspondence.

4. ΔPQR

5. ΔABD

6. ΔKLM

P

Q R S

A

B C

D

K

L M

N

Day 59 Practice


7. ΔABF

8. ΔPSR

9. ΔJML

A

B

C D

F

P

Q R

S

K

J

L

M

Day 59 Practice


10. ΔWXY

In questions 11-14, state whether the triangles are congruent or not based on the information given for

each. If congruent, identify the congruence postulate used.

11.

12. ΔKLM

W

X

Z

Y

P

Q M L

K

R

S

T U

V

Day 59 Practice


13.

14.

Given that the two triangles are congruent, find the values of the unknown letters in questions 15-20.

P

Q

M L

K R

𝑡 + 4.5

2𝑎

100° 3𝑏 − 3 2𝑐 + 4

112°

87°

3𝑑 2.4 in.

2𝑡 + 3

2𝑓 5.6 𝑖𝑛

Day 59 Practice


15. a

16. b

17. c

18. d

19. 𝑡

20. f

Day 59 Practice


Answer keys Day 59:

1. (In other marking that implies SSS can do)

2. (In other marking that implies SAS can do)

3. (In other marking that implies ASA can do)

P

Q M L

K

R

P

Q M L

K

R

P

Q M L

K

R

Day 59 Practice


4. ΔPSR

5. ΔCDB

6. ΔNML

7. ΔDBC

8. ΔRPQ

9. ΔJKL

10. ΔWZY

11. Congruent. S.A.A

12. Not congruent

13. Congruent S.S.S.

14. Congruent A.S.A

15. 𝑡 = 56°

16. 𝑏 = 30°

17.𝑐 = 48°

18. 0.8 in.

19. 1.5 in.

20. 2.8 in.

Day 59 Exit Slip


Use the figure below to answer the following questions.

Given that AB̅̅ ̅̅ ≅ ED̅̅ ̅̅ ; AC̅̅̅̅ ≅ EC̅̅̅̅ and ∠A ≅ ∠E.

(a) Name the triangle which is congruent to ΔEDC.

(b) Using the information given above, which triangle congruence postulate has been used to show that

the two triangles are congruent?

A

B C D

E

Day 59 Exit Slip


Answer Keys

Day 59:

(a) ΔABC

(b) S.A.S postulate

197


Geometry


Week 12


198

Week 12

Weekly Assessments

199

Week #12 1. Use the triangles below to answer the

questions that follow.

a) Which side corresponds to ST?

b) Which side corresponds to TU?

2. Identify the fixed points in the following

transformations.

a) A square reflected over its diagonal. b) A rectangle translated to the left by two units.

3. Use the diagram below to answer the questions

that follow. C is the midpoint of BD.

a) Which corresponds to BC? b) Show that ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶.

4. State whether the pairs of triangles are

congruent or not.

a) a)

60°

85° 35° S

T

U

60°

M

85°

N

O

A

B

C

D

2.2 𝑖𝑛 2.1 𝑖𝑛

200

5. Identify the postulate making each pair of the

triangles congruent.

a) b)

6. The triangles below are congruent.

a) Which angle corresponds to ∠𝐷? b) Which angle corresponds to ∠𝐹?

D E

F

Q

S S

201

Week 12 - Keys

Weekly Assessments

202

Week #12 KEY 1. Use the triangles below to answer the


a) Which side corresponds to ST?

MN

b) Which side corresponds to TU?

𝑀𝑂

2. Identify the fixed points in the following

transformations.

a) A square reflected over its diagonal. All the points on the diagonal b) A rectangle translated to the left by two units. No fixed points

3. Use the diagram below to answer the questions

that follow. C is the midpoint of BD.

a) Which side corresponds to BC? DC b) Show that ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶. 𝐴𝐵 ≅ 𝐵𝐶 (Given) AC is shared by the two triangles. Thus 𝐴𝐶 ≅ 𝐴𝐶 C is the midpoint of BD thus 𝐵𝐶 ≅ 𝐷𝐶. Therefore SSS postulate is achieved hence ∆𝐴𝐵𝐶 ≅ 𝐴𝐷𝐶

4. State whether the pairs of triangles are

congruent or not.

a) They are congruent a) They are not congruent

60°

85° 35° S

T

U

60°

M

85°

N

O

A

B

C

D

2.2 𝑖𝑛 2.1 𝑖𝑛

203

5. Identify the postulate making each pair of the

triangles congruent.

a) SAS postulate b) SSS postualate

6. The triangles below are congruent.

a) Which angle corresponds to ∠𝐷? ∠𝑆 b) Which angle corresponds to ∠𝐹?

∠𝑄

D E

F

Q

S S

Day 61 Bellringer


1.Identify the rigid motion that can map the pairs of triangles below.

a)

b)

Day 61 Bellringer


c)

2. State whether the following statements are true or force.

a) All rigid motions preserves the shape and the size of the figure?

b) Rigid motions changes the angle measure between two adjacent sides of a triangle.

Day 61 Bellringer


Answer Key Day 61

1. a) Reflection

b) Glide reflection

c) Translation

2. a) True

b) False

Day 61 Activity


1. Draw the diagonals of the plane paper as shown.

2. Label the plane paper as shown below.

3. Cut the rectangle through the diagonals such that you have four triangles.

4. Get ∆𝐾𝐷𝐿 and ∆𝐹𝐸𝐺 and adjust them until they coincide, (if possible).

Is ∆𝐾𝐷𝐿 ≅ ∆𝐹𝐸𝐺?

5. Move ∆𝐴𝐵𝐶 and ∆𝐻𝐼𝐽 and adjust them until they coincide (if possible)

Is ∆𝐴𝐵𝐶 ≅ ∆𝐻𝐼𝐽?

6. Move ∆𝐸𝐹𝐺 and ∆𝐻𝐼𝐽 closer to see if they coincide, (if possible). Is ∆𝐸𝐹𝐺 ≅ ∆𝐻𝐼𝐽?

7. Move ∆𝐾𝐷𝐿 and ∆𝐻𝐼𝐽 closer to see if they coincide, (if possible). Is ∆𝐾𝐷𝐿 ≅ ∆𝐻𝐼𝐽?

8. Among the triangles that are congruent, Identify the transformation that would map one on another

as they appear in their original position in the rectangle.

A B

D E H

I J

F

G

K

L

Day 61 Activity


In this activity students will cut different triangles out of a rectangular plane paper and identify

congruent and non-congruent triangles. Students will work in groups of at least three and each group is

required to have a plane paper, a ruler, a pencil and a razor or anything that can cut a piece of paper.

Answer Keys

Day 61:

1-3. No response

4. Yes

5. Yes

6. No

7. No

8. Reflection about a line parallel to the width or the length

Or rotation about the point which is the intersection of the two diagonals of the rectangle

Day 61 Practice


Use the graph below to answer questions 1 and 2.

1. Identify a rigid motion that will map ∆𝐴𝐵𝐶𝐷 onto ∆𝐴′𝐵′𝐶′𝐷′.

2. Is ∆𝐴𝐵𝐶𝐷 ≅ ∆𝐴′𝐵′𝐶′𝐷′?

Use the diagram below to answer question 3 and 4.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4 A B

C

𝐶′

𝐵′ 𝐴′

D

𝐷′

Day 61 Practice


3. Which rigid motion will map the two triangles above?

4. Are the two triangles congruent to each other?

Use the diagram below to answer questions 5 and 6.

5. Identify a rigid motion that will map one triangle onto the other.



7. Identify a rigid motion that can map the two triangles.

8. Are the two triangles congruent to each other.

Day 61 Practice


Study the graph below and use it to answer questions 9 and 10.

9. Identify a rigid motion that can map ∆𝐴𝐵𝐶 onto ∆𝐴′𝐵′𝐶′.

10. Is ∆𝐴𝐵𝐶 ≅ ∆𝐴′𝐵′𝐶′?

Use the diagram below to answer questions 11 and12.

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4

A B

C 𝐶′

𝐵′ 𝐴′

Day 61 Practice


11. Which rigid motion will map one triangles onto the other?




14. Are the triangles congruent to each other?

Use the graph below to answer questions 15 and 16.

Day 61 Practice


15. Which rigid motion will map ∆𝐴𝐵𝐶 to ∆𝑀𝑁𝑂?

16. Is ∆𝐴𝐵𝐶 ≅ ∆𝑀𝑁𝑂







Day 61 Practice


Answer Keys

Day 61:

1. A rotation of 180° or −180° about the origin.

2. Yes

3. None

4. No

5. Glide reflection

6. Yes

7. Translation

8. Yes

9. Reflection

10. Yes

11. None

12. No

13. Translation

14. Yes

15. A rotation of +90° about the origin

16. Yes

17. Reflection

18. Yes


20. Yes

Day 61 Exit Slip


1. Is ∆𝐴𝐵𝐶 ≅ ∆𝑆𝑇𝑈? Explain

A B

C

S T

U

62°

62°

Day 61 Exit Slip


Answer Keys

Day 61:

Yes;

The figures are related by glide reflection which a rigid motion

Day 62 Bellringer


Given that ΔPQR and ΔXYZ below are congruent, identify the angle or side indicated in each case. Use

the correct correspondence when naming the angles and sides.

(a) PR̅̅̅̅

(b) ∠QPR

(c) YZ̅̅̅̅

(d) ∠YZX

(e) XZ̅̅̅̅

P

Q

R

X

Y

Z

Day 62 Bellringer


Answer Keys Day 62:

(a) YZ̅̅̅̅

(b) ∠XYZ

(c) PR̅̅̅̅

(d) ∠PRQ

(e) QR̅̅ ̅̅

Day 62 Activity


1. Measure the lengths of 𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ and 𝐴𝐶̅̅ ̅̅ in inches using a ruler and write down the measurements.

2. Find the measures of ∠𝐴, ∠𝐵 and ∠𝐶 using a protractor and write down these measures.

3. Using a tracing paper and a ruler, carefully duplicate ΔABC and label it ΔKLM in the order shown

below. Are these triangles congruent?

4. Measure the length of KL̅̅̅̅ on ΔKLM using a ruler and write it down. Which side on ΔABC has the

same length as KL̅̅̅̅ ?

A B

C

K L

M

Day 62 Activity


5. Measure the length of LM̅̅ ̅̅ on ΔKLM using a ruler and write it down. Which side on ΔABC has the

same length as LM̅̅ ̅̅ ?

6. Similarly, measure the length of KM̅̅̅̅̅ on ΔKLM using a ruler and write it down. Which side on ΔABC

has the same length as KM̅̅̅̅̅?

7. Find the measure of ∠𝐾 in ΔKLM using a protractor and write down its measure. Which angle in

ΔABC has the same measure as ∠𝐾?

8. Find the measure of ∠𝐿 in ΔKLM using a protractor and write down its measure. Which angle in

ΔABC has the same measure as ∠𝐿?

9. Similarly, find the measure of ∠𝑀 in ΔKLM using a protractor and write down its measure. Which

angle in ΔABC has the same measure as ∠𝑀?

Day 62 Activity


In this activity students will learn how to identify congruent angles and sides using congruent triangles.

Students will work in groups of two or four and each group will require a tracing paper of A4 size, a ruler

and a protractor. The students in the respective groups will also be provided with a copy of ΔABC shown

below.

Answer keys

Day 62:

1. Lengths should be measured accurately

2. Angles should be measured accurately

3. Yes

4. 𝐴𝐵̅̅ ̅̅

5. 𝐵𝐶̅̅ ̅̅

6. 𝐴𝐶̅̅̅̅

7. ∠𝐴

8. ∠𝐵

9. ∠𝐶

Day 62 Practice


Use the figures below to answer questions 1-6.

Given that ΔABC ≅ ΔPQR, identify the parts congruent to the ones indicated in order to show

congruence between the two triangles. Use the correct correspondence in each case.

1. 𝐴𝐶̅̅̅̅

2. ∠𝑄

3. 𝑃𝑄̅̅ ̅̅

4. ∠𝐵

5. 𝐵𝐶̅̅ ̅̅

6. ∠𝑅

A

B

C

P

Q

R

Day 62 Practice


Use the figures below to answer questions 7-15.

Triangle KLM is congruent to triangle XYZ.

Identify the parts congruent to the following parts:

7. 𝐾𝐿̅̅̅̅

8. ∠𝑋

9. 𝑋𝑌̅̅ ̅̅

10. ∠𝐿

11. 𝐿𝑀̅̅ ̅̅

12. ∠𝑌

K

L

M

X

Y

Z

62°

70°

48°

Day 62 Practice


Find the values represented by the following letters:

13. 𝑥

14. 𝑦

15. 𝑧

Use the triangles to answer questions 16-20.

Δ𝐹𝐺𝐻 and Δ𝐽𝐾𝐿 shown below are congruent. The triangles are not drawn to scale.

Identify parts congruent to the following ones:

16. ∠J

F G

H

J

K

L

4 𝑖𝑛.

2 + 𝑏

8 𝑖𝑛.

2𝑎 − 4

Day 62 Practice


17. 𝐹𝐻̅̅ ̅̅

18. ∠K

Hence find the values represented by the letters:

19. 𝑎

20. 𝑏

Day 62 Practice


Answer keys

Day 62:

1. 𝑄𝑅̅̅ ̅̅

2. ∠𝐴

3. 𝐵𝐴̅̅ ̅̅

4. ∠𝑃

5. 𝑃𝑅̅̅ ̅̅

6. ∠𝐶

7. 𝑌𝑍̅̅̅̅

8. ∠𝑀

9. 𝑀𝐾̅̅ ̅̅ ̅

10. ∠𝑍

11. 𝑍𝑋̅̅ ̅̅

12. ∠𝐾

13. 𝑥 = 25°

14. 𝑦 = 26°

15. 𝑧 = 5°

16. ∠𝐻

17. 𝐾𝐽̅̅ ̅

18. ∠𝐹

19. 𝑎 = 6

20. 𝑏 = 2

Day 62 Exit Slip


In the figure below AB ∥ CD and AB ≅ CD.

Identify the side that is congruent to OC̅̅̅̅ in ΔCOD and ∠ABO in ΔAOB.

A

B

C

D

O

Day 62 Exit Slip


Answer Keys

Day 62:

OA̅̅ ̅̅ and ∠CDO

Day 63 Bellringer


The diagram below shows a pre-image and its image under a transformation.

1. What is common among lines JT, GU and HV?

2.Identify common features among the two diagrams

3. Identify the transformation mapping one image to another


5. Are the two diagrams congruent? Explain.

J

H

G

T

V

U

Day 63 Bellringer


Answer Keys

Day 63:

1. They are parallel and equal

2. Have the same orientation and size

3. Translation

4. Respective image point moves through the same distance and direction

5. Yes, the image is mapped to the object by a translation which is a rigid motion. The fact that

rigid motion preserves distance and angular measurement, the figures are congruent.

Day 63 Activity


1. On the graph paper using a suitable scale draw two horizontal lines segments of 3 – 4 units. Let the lie

on one horizontal line of a graph paper.

2. Label the lines as XY and MN respectively.

3. On the right end of each line, draw a vertical line of about 5 – 7 units so that they are vertical to the

lines XY and MN above. Label the endpoints as Z and P so that YZ and NP are perpendicular to XY and

MN respectively.

4. Connect the lines so that we have two triangles XYZ and MNP are two triangles.

5. Measure XZ and MP, compare the measurements.

6. Confirm that the corresponding angles are congruent by first, measuring the corresponding angles

and comparing them.

7. Find the area of each triangle.

8. Compare the size of the triangles, what do you notice.

Day 63 Activity


In this activity, we would like to establish that if corresponding angles and sides of a triangle congruent,

then the two triangles are congruent. Student will work in groups of at least 3. Each group will require a

ruler, a pencil, and a graph paper.

Answer Keys Day 63:


5. Different responses. The two measurements should be equal

6. Angle measurements will vary from group to group, however, the corresponding angles must be

approximately equal

7. Different responses but equal area in each group


Day 63 Practice


Use the following information to answer questions 1 - 8

1. Identify corresponding angles in the above figure

2. Identify corresponding sides in the above figure

3. Find the relationship between the corresponding sides

4. Identify the transformation between the two figures above.

5. Are two figures congruent?

6. Explain your answer 5 above.

7. Having established 5 above, what could be the relationship between the corresponding angles?

8. Explain why the relationship in 7 above.

Q

K

L

Y

X

H

8 in

8 in

6.5 in

6.5 in

4 in

4 in

Day 63 Practice


Use the following information to answer the following questions

9. Identify corresponding angles

10. Identify corresponding sides

11. Find the size of angle EDC.

12. Find the size of angle FHG, state the reason for your answer.

13. Find the relationship between the corresponding angles

14. Explain what the answer in 13 suggest.

15. Find the relationships between the corresponding sides.

C

D

E

F

G

H

9.5 in 87°

5.4 in

55°

38°

87°

5.5 in

Day 63 Practice


16. Combining the answer in 15 and 13, what is the relationship between the two triangles?


Use the information to answer questions 18 – 20.

Two triangles are related by a dilation of scale factor –1.

18. Are the two triangles congruent?


20. Based on your answer in 18 above, what is the consequences using the theorem

Day 63 Practice


Answer keys Day 63:

1. ∠𝐾 and ∠𝐻, ∠𝑄 and ∠𝑌, ∠𝐿 and ∠𝑋.

2. 𝐾𝑄 and 𝑌𝐻, 𝐾𝐿 and 𝐻𝑋, 𝐿𝑄 and 𝑋𝑌

3. 𝐾𝑄 = 𝑌𝐻 = 4 𝑖𝑛, 𝐾𝐿 = 𝐻𝑋 = 6.5 𝑖𝑛 𝐿𝑄 = 𝑋𝑌 = 8 𝑖𝑛

4. Rotation

5. Yes

6. They are mapped onto each other, by rotation, a rigid motion which of course preserves length and

angle measurement, thus, leading to congruent image and pre-image.


8. From the theorem on congruence, when the two triangles are congruent, their corresponding angles

are too.

9. ∠𝐸 and ∠𝐻, ∠𝐶 and ∠𝐺, ∠𝐷 and ∠𝐹.

10. 𝐶𝐸 and 𝐺𝐻, 𝐸𝐷 and 𝐺𝐹, 𝐷𝐶 and 𝐹𝐻

11. 𝐴𝑛𝑔𝑙𝑒 𝐸𝐷𝐶 = 38°

12. 𝐴𝑛𝑔𝑙𝑒 𝐹𝐻𝐺 = 55°

13. ∠𝐸 = ∠𝐻, ∠𝐶 = ∠𝐺, ∠𝐷 = ∠𝐹.

14. Based on comparison of the angle measures given and those computed

15. No sufficient information given, hence not related but, at least a pair or corresponding angles are

not equal

16. They are not congruent

17. Corresponding angles are equal but corresponding sides are not equal hence, we cannot apply the

theorem on congruence of triangles. Thus, the triangles are not congruent.

18. YES

19. Have a scale factor of absolute one and corresponding angles are equal. This implies that length and

angular measure are maintained.

20. Corresponding sides and angles are equal.

Day 63 Exit Slip


Given that the two triangles are congruent, find the relation between their interior angles.

M

N P

S

T

U

5 in

5 in

Day 63 Exit Slip


Answer Keys

Day 63:

∠𝑁 = ∠𝑇, ∠𝑃 = ∠𝑈 𝑎𝑛𝑑 ∠𝑀 = ∠𝑆

Day 64 Bellringer


1.Identify the postulate that make the following triangles congruent.

a)

b)

c)

Day 64 Bellringer


2. In the figure below, ∆𝐴𝐵𝐶 is mapped onto ∆𝑀𝑁𝑂 by a reflection. Use the figure to answer the


a) Which side corresponds to side 𝐴𝐶?

b) Which angle corresponds ∠𝐴𝐵𝐶?

A B M N

C O

Day 64 Bellringer


Answer Key

Day 64

1. a) A.S.A postulate

b) S.A.S Postulate

c) S.S.S Postulate

2. a) NO

b) ∠𝑁𝑀𝑂

Day 64 Activity


1. Label the square as WXYZ as shown below.

2. Fold the square along the diagonal DB.

3. How many triangles are formed?

4. To they overlap?

5. What is the conclusion about their relation with respect to your answer in 4 above.

6. Is there a transformation relating the two triangles, if yes, which one.

7. Now fold it along AC.

8. Together with the first fold, how many triangles are formed.

9. List the triangles that are congruent to each other given that the diagonals meet at O.

10. What are the postulates used to come up with answers in 9 above

A B

C D

Day 64 Activity


In this activity, students will determine the congruent triangles in a square when divided by its

diagonals. They will do so using the postulates. Students will work in groups of at least 3 where each

group will require a square shaped plain paper, a ruler and a pencil.

Answer Keys

Day 64:


3. 2

4. They do


6. Reflection along DB

7. No response

8. 8

∆𝐷𝐶𝑂, ∆𝐶𝐵𝑂, ∆𝐵𝐴𝑂 𝑎𝑛𝑑 ∆𝐴𝐷𝑂

∆𝐴𝐵𝐶, ∆𝐴𝐷𝐶, ∆𝐵𝐷𝐴 𝑎𝑛𝑑 ∆𝐵𝐷𝐶

9.SSS or SAS or ASA.

Day 64 Practice


Use the diagram below to answer questions 1 - 4.

1. Which postulate show that the triangles are congruent?

2. Find the value of t.

3. Find the value of y

4. Find the value of 𝑥

Use the diagram below to answer questions 5 - 10.

AC and BD are straight lines and point O divides line AC in the ratio 2:1. Line AC is 9𝑖𝑛 long.

5. Which postulate shows that ∆𝐴𝑂𝐷 and ∆𝐶𝑂𝐵 are congruent?

6. Is ∆𝐴𝑂𝐵 ≅ ∆𝐶𝑂𝐷?

7. What is the length of DO?

(2𝑦 + 2)𝑖𝑛

(3𝑥 − 2)𝑖𝑛

(𝑥 + 4)𝑖𝑛

(4𝑦 − 2)𝑖𝑛

(70 + 𝑡)° (3𝑡 − 80)°

O

A B

C D

Day 64 Practice


8. What is the length of OB?

9. Is ∆𝐴𝐶𝐷 ≅ 𝐵𝐶𝐷?

10. If yes, which postulate is used in 9 above. If no, Identify the condition that is not met by the

postulate.

Use the rectangle below to answer questions 11 - 13.

11. Which triangle is congruent to ∆𝑆𝑂𝑇?

12. Which triangle is congruent to ∆𝑇𝑂𝑈?

13. Is ∆𝑇𝑂𝑈 ≅ ∆𝑆𝑂𝑇?

Use the figure below to answer questions 14 - 16.

14. Which postulate makes the two triangles congruent?

Day 64 Practice


15. What is the value of a?

16. What is the value of 𝑥?

17. Find the value of c?

From question 18 - 20, state the postulate which makes each pair of triangles congruent.

18.

19.

20.

Day 64 Practice


Answer Key

Day 64:

1. S.A.S

2. 75

3. 2

4. 3

5. S.A.S

6. No

7. 3𝑖𝑛

8. 6𝑖𝑛

9. Yes

10. S.S.S

11. ∆𝑈𝑂𝑉

12. ∆𝑆𝑂𝑉

13. No

14. A.S.A

15. 4 in

16. 35°

17. 53

18. S.S.S

19. S.A.S

20. A.S.A

Day 64 Exit Slip


1. Is ∆𝑀𝑁𝑂 ≅ 𝑆𝑇𝑈? Explain

𝑀 𝑁

O

S T

U

Day 64 Exit Slip


Answer Keys

Day 64:

1. Yes

The corresponding sides are equal, thus, we have SSS postulate

Unit 4 Test Name ____________________________________


Questions:

1. Identify the rigid motion involved in the following transformation.

2. A square of sides 5 cm by 5 cm was translated three units upwards and six units to the right. What

is the area of the image?

3. What is the value of a?

Unit 4 Test Name ____________________________________


4. Define a rigid motion!

5. Is this kind of transformation a rigid motion or not? Why?

6. Describe a single rigid motion which can map ∆𝐴𝐵𝐶 to its image.

Unit 4 Test Name ____________________________________


7. Identify the congruent sides.

8. Which type of rigid motion has mapped ΔPQR onto ΔXYZ?

Unit 4 Test Name ____________________________________


9. ΔPQR has been mapped onto ΔABC after a half turn about the origin (0,0). Considering ΔPQR and

ΔABC, identify all the pair of congruent angles.

10. State whether each of the following pairs of triangles is congruent or not.

a)

Unit 4 Test Name ____________________________________


b)

11. Find the value of x if these two triangles are congruent.

12. ∆𝐴𝐵′𝐶′ is an image of ∆𝐴𝐵𝐶. Find the center of dilation.

Unit 4 Test Name ____________________________________


13. The triangle is dilated about point P with a scale factor of -1. Indicate the image.

14. PQ ≅ PR and PT bisects QR. Show that the two triangles are congruent.

15. Find the third angle of a triangle whose first two angles are 33° and 67°.

16. Find the measure of angle A.

Unit 4 Test Name ____________________________________


17. State the congruence postulate that makes the following triangles congruent.

18. Define:

a) Congruent angles

b) Congruent sides

19. The diagram shows a pre-image and its image under a transformation. What is common among

lines JT, GU and HV?

Unit 4 Test Name ____________________________________


20. Is ∆𝑀𝑁𝑂 ≅ ∆𝑆𝑇𝑈? Why?

Unit 4 Test Name ____________________________________


Answers:

1. Translation

2. 25 𝑐𝑚2

3. 67°

4. A rigid motion is a transformation on a plane that preserves the distance between the points and

the relative position of the points. There are four rigid motions namely reflection, glide reflection,

translation and rotation.

5. It is not a rigid motion because the size has changed after the transformation.

6. A rotation of 180° about the point (0, 1).

7. YX and QS, XL and SE, and YL and QE

8. Reflection

9.

∠P ≅ ∠A

∠Q ≅ ∠C

∠R ≅ ∠B

10.

a) No

b) Yes

11. x=2 in

12. Point A

13.

14.

Unit 4 Test Name ____________________________________


15. 80°

16. 30 degrees

17. ASA postulate

18.

a) Congruent angles that have the same measure.

b) Congruent sides are sides that have the same length.

19. They are parallel and equal.

20. Yes. The corresponding sides are equal, thus, we have SSS postulate.

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