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Slide 2 / 209

www.njctl.org

CongruentTriangles

Geometry

2014-06-03

Slide 3 / 209

Table of ContentsClassifying TrianglesInterior Angle Theorems

Isosceles Triangle Theorem

Congruence & TrianglesSSS CongruenceSAS CongruenceASA CongruenceAAS CongruenceHL Congruence

CPCTCTriangle Coordinate Proofs

Triangle Congruence Proofs

Exterior Angle Theorems

Slide 4 / 209

Return to Tableof Contents

ClassifyingTriangles

Slide 5 / 209

Parts of a triangle

Side

Side

Vertex

Vertex VertexA B

C

Side opposite

andare adjacent sides

interior

Vertex (vertices) - points joining the sides of triangles

Adjacent Sides - two sides sharing a common vertex

Slide 6 / 209

Parts of a triangle (cont'd)

hypotenuseleg

leg

leg

leg

base

In a right triangle, the hypotenuse is the side opposite the right angle. The legs are the 2 sides that form the right angle.

In an isosceles triangle, the base is the side that is not congruent to the other two sides (legs).

If an isosceles triangle has 3 congruent sides, it is an equilateral triangle.

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Definitions

Triangle - three-sided polygon

Polygon - a closed plane figure composed of line segments

Sides - the line segments that make up a polygonVertex (vertices) - the endpoints of the sidesAcute Triangle - all angles < 90°

Obtuse triangle - one angle is between, 90° < angle < 180°

Right Triangle - one 90° angle

Equiangular Triangle - 3 congruent angles

Equilateral Triangle - 3 congruent sides

Scalene triangle - No congruent sides

Isosceles Triangle - 2 congruent sides

Slide 8 / 209A triangle is formed by line segments joining three noncollinear points. A triangle can be classified by its sides and angles.

Equilateral

3 congruent sides

Isosceles

2 congruent sides

Scalene

No congruent sides

Classification by Sides

Classification by Angles

3 acuteangles

Acute Equiangular

3 congruentangles

Right

1 right angle

Obtuse

1 obtuse angle

(also acute)

Slide 9 / 209Classify the triangles by sides and angles

equilateralequiangular

scaleneacute

isoscelesacute

isoscelesobtuse

isoscelesright

click

clickclick

click click

Slide 10 / 209

Measure and Classify the triangles by sides and anglesExample

isosceles, right isosceles, acuteClick for AnswerClick for Answer Click for AnswerClick for Answer scalene, obtuseClick for AnswerClick for Answer

Slide 11 / 209

Measure and Classify the triangles by sides and anglesExample

scalene, obtuse scalene, acuteClick for AnswerClick for Answerequilateral, acute/equiangularClick for AnswerClick for Answer Click for AnswerClick for Answer

Slide 12 / 209

1 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

(Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

[This object is a pull tab]

Ans

wer

C Scalene

Bonus: F Right32+42 = 52

9+16 = 25

Slide 13 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 13 (Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

B Isosceles

Bonus: D Acute

3 cm 3 cm

2 cm

Slide 14 / 209


A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse

Ans

wer



A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse


Ans

wer

A Equilateral

Bonus:E Equiangular (all equilateral triangles are equiangular)D Acute (all angles are 60o )

Slide 15 / 209

4 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

(Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

F Right

Bonus: C Scalene (all angles are different, so all sides are different)

Slide 16 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

G Obtuse

Bonus: C Scalene (all angles are different, so all sides are different)

Slide 17 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

D AcuteE Equiangular

Bonus:A Equilateral (if a triangle is equiangular, then it's equilateral)

Slide 18 / 209

7 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

(Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

C ScaleneF Right

Slide 19 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer A Equilateral

D AcuteE Equiangular

Slide 20 / 209

9 Classify the triangle by sides and angles

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

A B120°

C

Ans

wer



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

A B120°

C


Ans

wer

C ScaleneG Obtuse

Slide 21 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

L

MN

Ans

wer

(Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

L

MN


Ans

wer

B IsoscelesF Right

Slide 22 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse H

J

K45°

85°

50° Ans

wer



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse H

J

K45°

85°

50°


Ans

wer

C ScaleneD Acute

Slide 23 / 209

12 An isosceles triangle is _______________ an equilateral triangle.

A Sometimes

B Always

C Never

Ans

wer


12 An isosceles triangle is _______________ an equilateral triangle.

A Sometimes

B Always

C Never


Ans

wer

A Sometimes

Slide 24 / 209

13 An obtuse triangle is _______________ an isosceles triangle.

A Sometimes

B Always

C NeverA

nsw

er

(Answer) / 209

13 An obtuse triangle is _______________ an isosceles triangle.

A Sometimes

B Always

C Never


Ans

wer

A Sometimes

Slide 25 / 209

14 A triangle can have more than one obtuse angle.

True

False

Ans

wer


14 A triangle can have more than one obtuse angle.

True

False


Ans

wer

False

Slide 26 / 209

15 A triangle can have more than one right angle.

True

False

Ans

wer


15 A triangle can have more than one right angle.

True

False


Ans

wer

False

Slide 27 / 209

16 Each angle in an equiangular triangle measures 60°

True

False

Ans

wer

(Answer) / 209

16 Each angle in an equiangular triangle measures 60°

True

False


Ans

wer

True

Slide 28 / 209

17 An equilateral triangle is also an isosceles triangle

True

False

Ans

wer


17 An equilateral triangle is also an isosceles triangle

True

False


Ans

wer

False

Slide 29 / 209

InteriorAngle

Theorems


Slide 30 / 209

T1. Triangle Sum TheoremThe measures of the interior angles of a triangle sum to 180°

A B

C

If you have a triangle, then you know the sum of its three interior angles is 180°

Why is this true? Click here to go to the lab titled, "Triangle Sum Theorem"

Slide 31 / 209

Example: Triangle Sum TheoremFind the measure of the missing angle

J

K L

32°

20°

Theorem T1. The Triangle Sum Theorem says that the interior angles of must sum to 180°.

and substituting the information from the diagram

32° + + 20° = 180°

So,

+ 52° = 180°= 128° Check: 128+32+20=180

page2svg

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-sum-theorem-lab/

/ 209

A B

C

52°

53°

18 What is the measurement of the missing angle?

m∠B =

Ans

wer


A B

C

52°

53°


m∠B =


Ans

wer 52 + 53 + m<B = 180°

105 + m<B = 180° -105 -105 m<B = 75°

Slide 33 / 209


57°L

M

N

m∠N =

Ans

wer



57°L

M

N

m∠N =


Ans

wer 90 + 57 + m<N = 180°

147 + m<N = 180° -147 -147 m<N = 33°

Slide 34 / 209

47°

34°

x°

20 What is the measure of the missing angle?

x =

Ans

wer


47°

34°

x°

20 What is the measure of the missing angle?

x =


Ans

wer 34 + 47 + x = 180°

81 + x = 180° -81 -81 x = 99°

/ 209

(draw a diagram)

21 In ABC, if m B is 84° and m C is 36°, what is the m A?

Ans

wer


(draw a diagram)

21 In ABC, if m B is 84° and m C is 36°, what is the m A?


Ans

wer

triangle ABC = 180°So, 84 + 36 + = 180°

=180° - 120°= 60°

Slide 36 / 209

(draw a diagram)

22 In DEF, if m D is 63° and m E is 12°, what is the m F?

Ans

wer


(draw a diagram)

22 In DEF, if m D is 63° and m E is 12°, what is the m F?


Ans

wer triangle DEF = 180°

So, 63 + 12 + m F = 180°m F =180° - 75°m F = 105°

Slide 37 / 209

We can solve more "complicated" problems using the Triangle Sum Theorem.

Solve for x

55°

(12x+8)°

(8x-3)°P

Q

R

From the Triangle Sum Theorem

Example

55 + (12x+8) + (8x-3) = 180 Substituting from the diagram20x + 60 = 180 Combining like terms

20x = 120 Isolating x using inverse operationsx = 6

Slide 38 / 209

23 Solve for x in the diagram.

Q

R

S2x° 5x°

8x°

What is m Q m R m S

Extension

Click to reveal

Ans

wer

(Answer) / 209


Q

R

S2x° 5x°

8x°

What is m Q m R m S

Extension

Click to reveal


Ans

wer

2x+5x+8x = 180 15x = 180 x = 12

m Q = 24° m R = 96° m S = 60°

Extension Answer

Slide 39 / 209

Solve for x3x-17 +x+40 +2x-5 = 180°

24 What is the measure of angle B?

A

B

C

Hint

Click to reveal

Ans

wer


Solve for x3x-17 +x+40 +2x-5 = 180°

24 What is the measure of angle B?

A

B

C

Hint

Click to reveal


Ans

wer

3x-17+2x-5+x+40 = 180 6x + 18 = 180 6x = 162 x = 27 m<B = 3(27)-17 = 81 - 17 = 64

Slide 40 / 209

Corollary to Triangle Sum TheoremThe acute angles of a right triangle are complementary.

A B

C

Since T1. the Triangle Sum Theorem says the interior angles of a triangle must sum to 180°. So, 180° - 90° (the right angle) = 90° left between and .

Recall: two angles that add up to 90° are called complementary

Slide 41 / 209

Example

x°

5x°

The measure of one acute angle of a right triangle is five times the measure of the other acute angle.

Find the measure of each acute angle.

Since this is a right triangle, we can use the Corollary to the Triangle Sum Theorem which says the two acute angles are complementary. So,

x + 5x = 906x = 90x = 15

(using the Triangle Sum Theorem is a little more work)

One acute angle is 15° and the other is 75°

Slide 42 / 209

25 In a right triangle, the two acute angles sum to 90°

True

False

Ans

wer

(Answer) / 209

25 In a right triangle, the two acute angles sum to 90°

True

False


Ans

wer

True

Slide 43 / 209


57°L

M

N

Ans

wer



57°L

M

N


Ans

wer

x+57 = 90x = 33

Note: we solved this problem earlier using the Triangle Sum Theorem. Use the Corollary to the Triangle Sum this time.

Slide 44 / 209

What are the measures of the three angles?

27 Solve for x

ChallengeClick to reveal

A

B C

Ans

wer


What are the measures of the three angles?

27 Solve for x

ChallengeClick to reveal

A

B C


Ans

wer

3x-1+31 = 903x + 30 = 90 3x = 60 x = 20

Challenge Answerm<A = 59o

m<B = 90o

m<C = 31o

Slide 45 / 209

28 Solve for x

What are the measures of the three angles?Challenge

Click to reveal

D E

F

Ans

wer

(Answer) / 209

28 Solve for x

What are the measures of the three angles?Challenge

Click to reveal

D E

F [This object is a pull tab]

Ans

wer

2x-2+x+5 = 903x + 3 = 90 3x = 87 x = 29

Challenge Answerm<D = 90o

m<E = 56o

m<F = 34o

Slide 46 / 209

x°

2x°

G

H

J

29 In the right triangle given, what is the measurement of each acute angle?

Ans

wer


x°

2x°

G

H

J

29 In the right triangle given, what is the measurement of each acute angle?


Ans

wer

2x+x = 903x = 90x = 30

m<G = 30o

m<H = 60o

m<J = 90o

Slide 47 / 209

1

23

30 m 1 + m 2 = _______o

Ans

wer


1

23

30 m 1 + m 2 = _______o


Ans

wer

90

Slide 48 / 209

1

23

31 m 1 + m 3 = _________o

Ans

wer

(Answer) / 209

1

23

31 m 1 + m 3 = _________o


Ans

wer

90

Recall that <2 & <3 are vertical angles & vertical angles are congruent, so m<2=m<3 & therefore m<1+m<3 = 90

Slide 49 / 209

20°

X°

32 Find the value of x in the diagram

Mark your vertical angles!Hintclick to reveal

Ans

wer


20°

X°

32 Find the value of x in the diagram

Mark your vertical angles!Hintclick to reveal

Slide 50 / 209

Exterior Angle Theorems

Return to Table of Contents

Slide 51 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

Exterior angle

Exterior angle

Exterior angles are adjacent to the interior angles.

Exterior angles and interior angles together form a straight line.

The sum of an exterior angle and an interior angle is 180 degrees.

Slide 52 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR

The adjacent angles form a straight line so thesum of the two angle measures will be 180o

page2svg

/ 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR


Slide 54 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

Q

R


Slide 55 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR

The sum of an interior angle and an adjacent exterior angle is 180 degrees

The sum of the interior angles of a triangle add up to 180 degrees

1

P

R Q

m P + m Q + m R = 180o

m Q + m 1 = 180o1

P

R Q

Slide 56 / 209

The Exterior Angle Theorem says : m 1 = m P + m R

The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.

1

P

R QProof of the Exterior Angle TheoremWe know the following is true :

1. m P + m Q + m R = 180o

2. m Q + m 1 = 180o

This implies that m 1 = m P + m R and the Exterior Angle Theorem is proved true

Slide 57 / 209

Example: Using the Exterior Angle Theorem

140oXo

Xo

P

QR

What is the value of X ?

The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.

140o = xo + xo

140 = 2x

70 = x

Slide 58 / 209

ExampleSolve for x using the Exterior Angle Theorem

21°

34°x°

The Exterior Angle Theorem says that the exterior angle, marked x°, is equal to the two nonadjacent interior angles.

x = 21 + 34

y°

So, the exterior angle x = 55°

We also know what y is 125o ?What does x° + y° have to equal? 180o

click

click

/ 209

Example: What are w and x ? 75 + 50 + x = 180 125 + x = 180 -125 -125 x = 55o

w = 75 + 50 w = 125o

What does w + x equal? 125 + 55 = 180

75o

50owo Xo

Slide 60 / 209

124

3

m 4 = 131 m 3 = 53, fill in all the angles.

53o

49o

131o

78o 180o

127o 49o

131o

180o

127o53o

78o

Slide 61 / 209

33 Solve for the exterior angle, x.

x°60°

55°Y°

Ans

wer


33 Solve for the exterior angle, x.

x°60°

55°Y°


Ans

wer

1150

Slide 62 / 209

34 m 1 = 25 and m 4 = 83 Find m 3 = ?

A 25

B 50

C 58

D 83124

3

Ans

wer


34 m 1 = 25 and m 4 = 83 Find m 3 = ?

A 25

B 50

C 58

D 83124

3


Ans

wer

C

/ 209

Y°

35 Find the value of x using the Exterior Angles Theorem?

A 34

B 17

C 60

D 86

Ans

wer


Y°

35 Find the value of x using the Exterior Angles Theorem?

A 34

B 17

C 60

D 86 [This object is a pull tab]

Ans

wer

B

94 = 60 + 2x34 = 2x17 = x

0

Slide 64 / 209

Y°

36 Find the value of y in the figure below.

A 34

B 17

C 60

D 86 Ans

wer


Y°

36 Find the value of y in the figure below.

A 34

B 17

C 60

D 86


Ans

wer

D

Slide 65 / 209

37 Using the Exterior Angles Theorem, find the value

of x.

A100

B51

C46

D23

Y°

Ans

wer


37 Using the Exterior Angles Theorem, find the value

of x.

A100

B51

C46

D23

Y°


Ans

wer

D

100 = 2x +3 +51100 = 2x +54 46 = 2x 23 = x

/ 209

38 What is the value of Y?

A80

B40

C51

D100

Y°

Ans

wer


38 What is the value of Y?

A80

B40

C51

D100

Y°


Ans

wer

A

Slide 67 / 209

39 Find the value of x.

A40

B37.5

C20

D10

(3x - 5)°

(x + 2)° 33° Ans

wer



A40

B37.5

C20

D10

(3x - 5)°

(x + 2)° 33°


Ans

wer

C

3x - 5 = (x + 2) + 333x - 5 = x + 35 2x = 40 x = 20

Slide 68 / 209

25 o

115o

P

S

R Tw

40 PS bisects RST , what is the value of w?

A100

B110

C115

D125

Ans

wer


25 o

115o

P

S

R Tw

40 PS bisects RST , what is the value of w?

A100

B110

C115

D125


Ans

wer

C

25 o

115o

PR T

25o

65o

S

/ 209

ExampleFind the missing angles in the diagram.

Teac

her N

ote


ExampleFind the missing angles in the diagram.


Teac

her N

ote

Find the measures of all angles togetherm<1 = 45o

m<2 = 90o

m<3 = 60o

m<4 = 60o

m<5 = 77o

m<6 = 77o

m<7 = 43o

Slide 70 / 209

41 Find the measure of angle 1.

40o

1

24 53

60o

Ans

wer



40o

1

24 53

60o


Ans

wer

40 + m<1 = 90o

m<1 = 50o

Slide 71 / 209


40o

1

24 53

60o

Ans

wer



40o

1

24 53

60o


Ans

wer

40 + m<2 = 180o

m<2 = 140o

/ 209

43

40o

1

24 53

60o

Ans

wer

Find the measure of angle 3.


43

40o

1

24 53

60o



Ans

wer

40 + m<3 = 180o

m<3 = 140o

Slide 73 / 209

44

40o

1

24 53

60o

Find the measure of angle 4.A

nsw

er


44

40o

1

24 53

60o



Ans

wer m<4 = 40o

vertical angles are congruent

Slide 74 / 209

45

40o

1

24 53

60o


Ans

wer


45

40o

1

24 53

60o



Ans

wer

m<4 = 40o

Interior angles of a triangle add up to 180, so40+60+m<5 = 180100+m<5 = 180m<5 = 80o

/ 209

Isosceles TriangleTheorem


Slide 76 / 209

Parts of an Isosceles TriangleAn isosceles triangle has at least two congruent sides (an equilateral triangle is an isosceles triangle w/three congruent sides)

If an isosceles triangle has exactly two congruent sides, the: - two congruent sides are called legs, - the noncongruent side is called the base, - the two angles adjacent to the base are the base angles,

leg leg

baseangles

vertex angle

base

The vertex angle is the angle opposite the base ORit is the angle included by the legs

Slide 77 / 209

T3. Base Angles Theorem (BAT)If two sides of a triangle are congruent, the angles opposite them are congruent.

If , then

A

B C

Corollary to BAT (T3)

If a triangle is equilateral, then it is equiangular.

A

B C

Slide 78 / 209

x°

y°

44°

Examples:Find the values of x & y in the isosceles triangle below.

x = 44; Base Angles are Congruent

y + 44 + 44 = 180; Triangle Sum Th.y + 88 = 180y = 92

Find the values of x & y in the isosceles triangle below.

x° y°

52° x = y; Base Angles are Congruent

x + y + 52 = 180; Triangle Sum Th.x + x + 52 = 180; Substitution2x + 52 = 1802x = 128x = 64

Slide 79 / 209

x°

y°

35°

46 Solve for the measurements of the angles x and y

Ans

wer


x°

y°

35°

46 Solve for the measurements of the angles x and y


Ans

wer x = 35

y = 110

page2svg

/ 209

x°

y°

72°

47 Solve for x and y.

Ans

wer


x°

y°

72°

47 Solve for x and y.


Ans

wer x = 36

y = 72

Slide 81 / 209

70°

48 What are the measurements of the base angles?A

nsw

er


70°

48 What are the measurements of the base angles?


Ans

wer

70 + 2x = 1802x = 110x = 55

Each base angle is 55o

70°

x° x°

Slide 82 / 209

49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?

A 71° B 38° C 83° D 104°

Ans

wer


49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?

A 71° B 38° C 83° D 104°


Ans

wer

A

/ 209

T4. Converse of the Base Angles TheoremIf two angles of a triangle are congruent, then the sides opposite them are congruent.

If , then

A

B C

Corollary to Converse of the BAT (T4)

If a triangle is equiangular, then it is equilateral.

A

B C

Slide 84 / 209

D

EF 4

50 What is the measurement of FD?

Ans

wer


D

EF 4

50 What is the measurement of FD?


Ans

wer

FD = 4

Triangle is equiangular, so it's equilateral too

Slide 85 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

A

BC

7

40o

Ans

wer



A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

A

BC

7

40o


Ans

wer

B & E

Slide 86 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G rightA

B

C

4

4

4

Ans

wer

(Answer) / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G rightA

B

C

4

4

4[This object is a pull tab]

Ans

wer

A, B, D & E

Slide 87 / 209

A

B C5

3 3113o


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

Ans

wer


A

B C5

3 3113o


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right


Ans

wer

B & F

Slide 88 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

12

12

Ans

wer



A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

12

12[This object is a pull tab]

Ans

wer

A, B, D & E

Slide 89 / 209

ExampleFind the value of x and y

x°

y°

1. First, consider the top triangle. The 3 marks indicate this is an equilateral triangle

2. From the Corollary to the BAT(T3), we know that an equilateral triangle is also equiangular

3. Since the Triangle Sum Theorem (T1) says the interior angles must sum to 180°, y° = 60.

x°

y°

/ 209

x°

60°

60°

60°

120°

4. Two adjacent angles whose non-shared sides form a straight line are a linear pair.

5. The supplement to 60° is 120° (60° + 120° = 180°)

6. Using the Base Angles Theorem (T3) and the Triangle Sum theorem (T1), we can determine x°

x°

60°

60°

60°

120°x°

x + x + 120 = 180 2x + 120 = 180 2x = 60 x = 30

Slide 91 / 209

55 What is the value of y?

A 120°

B 70°

C 55°

D 125°

70°

y°

Ans

wer


55 What is the value of y?

A 120°

B 70°

C 55°

D 125°

70°

y°


Ans

wer

D

Slide 92 / 209

50° x°

56 What is the value of x?

A 50°

B 25°

C 30°

D 130°

Ans

wer


50° x°

56 What is the value of x?

A 50°

B 25°

C 30°

D 130°


Ans

wer

B

Slide 93 / 209


A 3 2/3B 14

C 15

D 16

3x - 17

28 Ans

wer

(Answer) / 209


A 3 2/3B 14

C 15

D 16

3x - 17

28


Ans

wer

C

Slide 94 / 209

Congruence &

Triangles


Slide 95 / 209

CongruenceTwo figures are congruent if they have the exact size and shape (They are similar if they have the same shape, but a different size)

Congruent figures have a correspondence between their angles and sides where pairs of corresponding angles are congruent and pairs of corresponding sides are congruent.

A

B

C N

O

P

A

B

C N

O

P

Slide 96 / 209

ExampleThe two triangles are congruent , write: 1) a congruence statement 2) identify all congruent corresponding parts

A

B

C

D

E

F

Answer

Slide 97 / 209

A

B

C

D

E

F

Slide 98 / 209

Part Corresponding Side Corresponding Angle

A

B

D

C

E

page2svg

page73svg

/ 209

Problem

Corresponding Sides Corresponding Angles

(If you need, draw a diagram)

Teac

her N

otes


Problem

Corresponding Sides Corresponding Angles

(If you need, draw a diagram)


Teac

her N

otes Corresponding Sides

AB EFBC FGAC EG

=~=~=~

Corresponding Angles <A <E <B <F <C <G

=~=~=~

Have students arrive at the answers as a class, or independently.

Slide 100 / 209

58 What is the corresponding part to J

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~

Ans

wer


58 What is the corresponding part to J

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~


Ans

wer

D

Slide 101 / 209

59 What is the corresponding part to Q

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~

Ans

wer


59 What is the corresponding part to Q

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~


Ans

wer

B

/ 209

60 What is the corresponding part to QP

A JL

B LK C KJ D PQ

J

K L R Q

P

JKL PQR=~

Ans

wer


60 What is the corresponding part to QP

A JL

B LK C KJ D PQ

J

K L R Q

P

JKL PQR=~


Ans

wer

C

Slide 103 / 209

61 Write a congruence statement for the two triangles

A

B

C

D

Z

X

CV

B

Ans

wer

BVC XCZ=~

XCB BCX=~

VBC ZXC=~

CBV CZX=~


61 Write a congruence statement for the two triangles

A

B

C

D

Z

X

CV

B

BVC XCZ=~

XCB BCX=~

VBC ZXC=~

CBV CZX=~


Ans

wer

C

Slide 104 / 209

Y

ZW

X

What else can be marked congruent?

62 Complete the congruence statement

A

B

C

D

XYZ =~

XWZ

ZWX

WXZ

ZXW

Ans

wer


Y

ZW

X

What else can be marked congruent?

62 Complete the congruence statement

A

B

C

D

XYZ =~

XWZ

ZWX

WXZ

ZXW


Ans

wer

B

XZ ZX<W <Y

=~=~

Extension Answer

/ 209

T5. Third Angles TheoremIf two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent.

Q

R

S

T

U

V

Can you give a reason for why this might be true?If the sum of the interior angles is 180 o and both sets of angles are the same, then the third angles will have the same measure. Example: m S = m V = 40o & m R = m U = 80o degrees, then m Q = m T = 60o.Click to reveal

Slide 106 / 209

ExampleFind the value of x.

A

B

C45° 75°

(2x+40)°W

X

Y

1) From the Third AngleTheorem (T5), we know m B = m Y

2) The m B is easy to find withthe Triangle Sum Theorem (T1),

3) Substitute to find x

Slide 107 / 209

S

Q

R48°117°

H

I

J

63 What is the measurement of J A

nsw

er


S

Q

R48°117°

H

I

J

63 What is the measurement of J


Ans

wer

48 + 117 + m<R = 180165 + m<R = 180m<R = 15o

m<J = m<R = 15o

Slide 108 / 209

I

J

K

80°

32°

P

Q

R(2x+14)°

64 Solve for x

Ans

wer


I

J

K

80°

32°

P

Q

R(2x+14)°

64 Solve for x


Ans

wer

80+32+m<I = 180112+m<I = 180m<I=68o

m<I = m<P68 = 2x+1454 = 2x27 = x

/ 209


62° 78°Q

R

S

(3x+10)°

C

B

A Ans

wer



62° 78°Q

R

S

(3x+10)°

C

B

A


Ans

wer

62+78+m<R = 180140 + m<R = 180m<R = 40o

m<B = m<R3x+10 = 403x = 30 x = 10

Slide 110 / 209

T6. Properties of Congruent Triangles

Reflexive Property of Congruent TrianglesEvery triangle is congruent to itself A

B

C A

B

C

Symmetric Properties of Congruent Triangles

A

B

C D

E

F A

B

CD

E

F

Transitive Property of Congruent Triangles

A

B

C D

E

F D

E

F J

K

L A

B

C J

K

L

Slide 111 / 209

SSS Congruence


Slide 112 / 209

From the Congruence and Triangles section, you learned that two triangles are congruent if the 3 corresponding pairs of sides and the 3 corresponding pairs of angles are congruent.

However, we do not always need all 6 pieces of information to prove 2 triangles congruent.

Slide 113 / 209

Postulate:

Side-Side-Side (SSS) CongruenceIf three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

A B

C

E

D

F

Click here to go to the lab titled, "Triangle Congruence SSS"

page2svg

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sss-lab/

/ 209

Example

A F

K

BGSolution:The congruence marks on the sides show that:

Slide 115 / 209

Example

F

GH

K

J

G

F

H

J

K

JJ

Slide 116 / 209

A

B

C H

J

K

You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ

66 The congruence statement is ABC = HJK

True

False

Hint

~A

nsw

er


A

B

C H

J

K

You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ

66 The congruence statement is ABC = HJK

True

False

Hint

~


Ans

wer

True

Slide 117 / 209

R

S

T U

67 SRT = SUT

True

False

~

Ans

wer


R

S

T U

67 SRT = SUT

True

False

~


Ans

wer True

SSS holds becauseST = ST ; Reflexive Property

/ 209

A

B C Q R

S

3

4

5 3

4

5

68 ABC = _____?

A QRS B SRQ

C ACB

D RSQ

Ans

wer

~


A

B C Q R

S

3

4

5 3

4

5

68 ABC = _____?

A QRS B SRQ

C ACB

D RSQ

~


Ans

wer

B

Slide 119 / 209

SAS Congruence


Slide 120 / 209

Included angle: the angle made by two lines with a common vertex

41 °

A

B

C

Included side: the side between two angles

C

D

E

Slide 121 / 209

Postulate:

Side-Angle-Side (SAS) CongruenceIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

F

P

B

A

C

Q

Click here to go to the lab titled, "Triangle Congruence SAS"

Slide 122 / 209

Example

1 2

L

M

P

N

OIs there any information you can fill in?

So, listing the corresponding congruent parts:

page2svg

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sas-lab/

/ 209

200

5 units

Why Not SSA? Move the side with the length of 2 units and create a triangle.

2 units

200

5 units

Can a different triangle be made than the first one made?

Ans

wer


200

5 units

Why Not SSA? Move the side with the length of 2 units and create a triangle.

2 units

200

5 units

Can a different triangle be made than the first one made?


Ans

wer

200

5 units

2 units

Two different shapes with same sides and length

200

5 units2 units

Slide 124 / 209

69 What is the included angle of the given sides of the triangle?

A J

B K

C L

Hint: Draw the triangle!

JKL, sides KL and JK

Ans

wer


69 What is the included angle of the given sides of the triangle?

A J

B K

C L

Hint: Draw the triangle!

JKL, sides KL and JK


Ans

wer

B

Slide 125 / 209

P

QR

S

TV4 4

5 5

100° 100°

70 List the congruent parts of the triangles below. Is PQR = STV?

Yes

No

~

Ans

wer


P

QR

S

TV4 4

5 5

100° 100°

70 List the congruent parts of the triangles below. Is PQR = STV?

Yes

No

~


Ans

wer

Yes, by SAS

PQ = ST<Q = <TRQ = VT

~~~

/ 209

F

GH

X

Y Z46° 46°

1010

77

Why?

71 Is FGH = XYZ by SAS?

Yes

No

Ans

wer

~


F

GH

X

Y Z46° 46°

1010

77

Why?

71 Is FGH = XYZ by SAS?

Yes

No

~


Ans

wer

No, the angles marked congruent are not the included angles in the triangles.

FG = XYHF = ZX<H = <Z

~~~

Slide 127 / 209

A B

C D

72 Using SAS, what information do you need to show ABC = DCB

A DBC = ACB B B = C

C ABD = DCA D ABC = DCB

~~

~

~

~

Ans

wer


A B

C D

72 Using SAS, what information do you need to show ABC = DCB

A DBC = ACB B B = C

C ABD = DCA D ABC = DCB

~~

~

~

~


Ans

wer

D

Slide 128 / 209

73 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer



A SSS

B SAS

C Not congruent


Ans

wer

B

/ 209


A SSS

B SAS

C Not congruent

Ans

wer



A SSS

B SAS

C Not congruent


Ans

wer

B

Slide 130 / 209


A SSS

B SAS

C Not congruent

Ans

wer



A SSS

B SAS

C Not congruent


Ans

wer

A

Slide 131 / 209


A SSS

B SAS

C Not congruent

Ans

wer



A SSS

B SAS

C Not congruent


Ans

wer

C

/ 209


A SSS

B SAS

C Not congruent

Ans

wer



A SSS

B SAS

C Not congruent


Ans

wer

C

Slide 133 / 209


A SSS

B SAS

C Not congruent45° 45°

12 12

Ans

wer



A SSS

B SAS

C Not congruent45° 45°

12 12


Ans

wer

B

Slide 134 / 209


ASA Congruence

Slide 135 / 209

Postulate:

Angle-Side-Angle (ASA) CongruenceIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

R

Q S

T

U

V

Click here to go to the lab titled, "Triangle Congruence ASA"

page2svg

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-asa-lab/

/ 209

ExampleE

FM

G

H90°90°8

8

Vertical anglesare congruent

Slide 137 / 209

First: what data is given to you?

Second: if it is not already marked, check and mark the diagram with that information,

Third: check your congruence postulates - what piece of information are you missing (side/angle) and where does it need to be for your chosen congruence?

Slide 138 / 209

W

X

Y

79 What is the included side for X and W?

A YX

B YW

C XW

Ans

wer


W

X

Y

79 What is the included side for X and W?

A YX

B YW

C XW


Ans

wer

C

Slide 139 / 209

W

X

Y

80 What is the included side for X and Y

A XW

B YX

C YW

Ans

wer


W

X

Y

80 What is the included side for X and Y

A XW

B YX

C YW


Ans

wer

B

/ 209

M

N

O

P

81 What piece of information do we need to have ASA congruence between the two triangles?

A

B

C

D

Ans

wer


M

N

O

P


A

B

C

D


Ans

wer

C

NP = PN is true by the Reflexive Property. If you mark the diagram to show the corresponding congruent parts, you can see NP = PN is needed

~

~

Slide 141 / 209

A

B

C

D


A

B

C

D

Ans

wer


A

B

C

D


A

B

C

D


Ans

wer

B is true by the Reflexive Property. Since BDC is a right angle, BDA must also be a right angle since they form a linear pair. The included segment we need is

Slide 142 / 209

E

F G

M

H

83 Why is ?

A ASA

B vertical angles

C included angles

D congruent

Ans

wer


E

F G

M

H

83 Why is ?

A ASA

B vertical angles

C included angles

D congruent


Ans

wer

B

Mark the congruent vertical angles when you see two lines intersect

/ 209


A SSS

B SAS

C ASA

D Not congruent

S

Q R

TU

Ans

wer



A SSS

B SAS

C ASA

D Not congruent

S

Q R

TU[This object is a pull tab]

Ans

wer A

Slide 144 / 209

When you have overlapping figures that share sides and/or angles, marking the diagram with the given information & pulling the triangles apart (when needed) makes it much easier to understand the problem.

Slide 145 / 209

85 What type of congruence exists between the two triangles?A SSS

B SAS

C ASA

D Not congruent

J

L

M

N

K

L

Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?

Hints:

Ans

wer

click to reveal

click to reveal

click to reveal


85 What type of congruence exists between the two triangles?A SSS

B SAS

C ASA

D Not congruent

J

L

M

N

K

L


Hints:click to reveal

click to reveal

click to reveal


Ans

wer B

Slide 146 / 209

A

B

C

Q R


A SSS

B SAS

C ASA

D Not congruent

Mark the diagram with the given information. Be careful you don't always use all information

Hint

Ans

wer

click to reveal

(Answer) / 209

A

B

C

Q R


A SSS

B SAS

C ASA

D Not congruent

Mark the diagram with the given information. Be careful you don't always use all information

Hint

click to reveal


Ans

wer C

Slide 147 / 209

C

B

Q R

A

B

Q R


A SSS

B SAS

C ASA

D Not congruent



click to reveal

click to reveal

Ans

wer


C

B

Q R

A

B

Q R


A SSS

B SAS

C ASA

D Not congruent



click to reveal

click to reveal[This object is a pull tab]

Ans

wer A

Slide 148 / 209

vertical


A SSSB SASC ASAD Not congruent

At the intersection of two lines you always have _____ angles.

Hint

ST

N

D

A

Ans

wer

Click to Reveal Click


vertical


A SSSB SASC ASAD Not congruent

At the intersection of two lines you always have _____ angles.

Hint

ST

N

D

A

Click to Reveal Click[This object is a pull tab]

Ans

wer

B - SAS

Slide 149 / 209


A SSS

B SAS C ASA D Not Congruent

Ans

wer

(Answer) / 209


A SSS

B SAS C ASA D Not Congruent


Ans

wer

We have two Sides and one Angle congruent, but they are not in the correct order.

D - Not Congruent

Slide 150 / 209


A SSS B SAS C ASA D Not Congruent

C

P M

S

A

Hint:

Mark the given information into your diagram. Identifying vertical angles plays an important part.

Ans

wer

click to reveal



A SSS B SAS C ASA D Not Congruent

C

P M

S

A

Hint:

Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal


Ans

wer

C - ASA

Slide 151 / 209

AAS Congruence


Slide 152 / 209

Theorem (T7):

Angle-Angle-Side (AAS) CongruenceIf two angles and the nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.

Q

R

S

T

U

V

Slide 153 / 209

Why is AAS a Theorem ?

Given two triangles:

AB

C

P R

K

75° 75°

65° 65°

12 12

The Triangle Sum Theorem (T1) allows us to find the measurement of the third angle in each triangle. 180°-(65°+75°)= 40°

AB

C

P R

K

75° 75°

65° 65°

12 1240°

Since AAS follows from ASA, AAS is a theorem rather than a postulate

page2svg

/ 209

Example

C

A

H

T

1) Mark your diagram:C

A

H

T

C

A

H

TSo, by AAS,

congruence statement?

Slide 155 / 209

91 What type of congruence exists, if any, between the two triangles?

A SSS

B SAS

C ASA

D AAS

E Not CongruentD

E

F

GH

Ans

werH



A SSS

B SAS

C ASA

D AAS

E Not CongruentD

E

F

GH

H


Ans

wer D - AAS.

What is the congruence statement?

Slide 156 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

A

B C Q

RS

Ans

wer



A SSS

B SAS

C ASA

D AAS

E Not Congruent

A

B C Q

RS


Ans

wer E Not congruent

We need an included (ASA) or nonincluded side (AAS), which we don't have

Slide 157 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer

(Answer) / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer C ASA

The two triangles share a common side which is congruent via the Reflexive Property.

Slide 158 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

Q

W

E

R

T

Ans

wer



A SSS

B SAS

C ASA

D AAS

E Not Congruent

Q

W

E

R

T[This object is a pull tab]

Ans

wer D AAS

The vertical angles are the key. The congruent side is nonincluded so it cannot be ASA.

Slide 159 / 209


A SSS

B SAS

C ASA

D AAS

E Not CongruentA

S

D

F

G

H

Ans

wer



A SSS

B SAS

C ASA

D AAS

E Not CongruentA

S

D

F

G

H


Ans

wer

B SAS Imagine you are walking around the figures, you must encounter the congruent parts in the correct order to use SAS congruence.

Slide 160 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent Ans

wer

(Answer) / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer D AAS

Q: Are there vertical angles in this diagram?

Slide 161 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

AB

C

D Ans

wer



A SSS

B SAS

C ASA

D AAS

E Not Congruent

AB

C

D


Ans

wer

D AAS Marking the bisected angle and the common side (reflexive), You can see AAS holds.

Slide 162 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer



A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer

E Not Congruent There is no AAA Congruence. AAA does make them similar (same shape), but the size may be different.

Slide 163 / 209

HL Congruence


page2svg

/ 209

Theorem (T8):

Hypotenuse-Leg (HL) CongruenceIf the hypotenuse and a leg of one right triangle are equal to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.

J

K L

M

N O

If you have a right triangle, make sure you check if HL applies

Slide 165 / 209

Why does HL Congruence work?Recall another theorem for right triangles:

c2 = a2 + b2 a

bc

Pythagorean Theorem:

If we know the lengths of two sides of a right triangle, we can solve for the length of the third side. HL Congruence theorem applies when the corresponding hypotenuse and one of the legs is congruent. When this is the case, the two right triangles are congruent.

A

B C E

FG

J

K L

M

N O

Slide 166 / 209

Example Are the two triangles congruent?

A B

C

R S

TThese are right triangles so let's try for HL congruence

A B

C

R S

T

Slide 167 / 209

Q

R S

X

Y Z

Mark the given on your diagram. Note that it is a right triangle.


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Given: QS = XZ RS = YZ

~~

Ans

wer

HintClick to reveal


Q

R S

X

Y Z

Mark the given on your diagram. Note that it is a right triangle.


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Given: QS = XZ RS = YZ

~~

HintClick to reveal


Ans

wer

EQ

R S

X

Y Z

QSR = XZYby HL congruence

~

Slide 168 / 209

If they are congruent what is the congruence statement?


A SSSB SASC ASAD AASE HLF Not congruent

L

M

N O

P

Q

Ans

wer

(Answer) / 209




L

M

N O

P

Q


Ans

wer C

LMN = OPQ~

Slide 169 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

A

B

CD

E

F

Ans

wer




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

A

B

CD

E

F


Ans

wer F

Slide 170 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

T

U

V W

X

Y

Ans

wer




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

T

U

V W

X

Y


Ans

wer

C

TUV = WXY~

Slide 171 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Q

W

EY

Ans

wer

(Answer) / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Q

W

EY


Ans

wer

D

QWY = EWY~

Slide 172 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

N

M

O J

K

L

Ans

wer




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

N

M

O J

K

L


Ans

wer E

OMN = JKL~

Slide 173 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H

Ans

wer




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H


Ans

wer

E

EFH = GFH~

Slide 174 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

HA

nsw

er

(Answer) / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H


Ans

wer B

EFH = GFH~

Slide 175 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

K F

B M

Ans

wer




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

K F

B M


Ans

wer A

KFB = MBF~

Slide 176 / 209

< POY and < UYO


P O

UY

What angles are congruent when parallel lines are cut by a transversal?


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Ans

wer

Click to Reveal


< POY and < UYO


P O

UY

What angles are congruent when parallel lines are cut by a transversal?


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Click to Reveal


Ans

wer D

POY = UYO~

Slide 177 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

O K

MJ

Ans

wer

(Answer) / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

O K

MJ


Ans

wer F

Slide 178 / 209




A S

XZ Ans

wer





A S

XZ


Ans

wer E

ASZ = XZS~

Slide 179 / 209

Triangle Congruence Proofs


Slide 180 / 209

Congruent Reasons Summary(Drag ones that don't work out of the chart. Then put HL where it would belong.)

SSS

ASSSSASAS

AASSAAASA

AAA

HL0

3

1

2

Slide 181 / 209Example

A F

K

BGSolution (two-column):

1) Given

2) SSS Postulate

AF = BG, FK = GK KA = KB~

~ ~1)

2) AFK = BGK~

Statements Reasons

Given: AF = BG, FK = GK & KA = KB~~ ~

page2svg

/ 209

Example F

GH

J

K

HF = HJ~

Given

FG = JK~

Given

H is the midpoint of GK.

Given

GH = KH~

Def. of midpoint

FGH = JKH~

SSS

Solution (flow proof):

Teac

her N

otes


Example F

GH

J

K

HF = HJ~

Given

FG = JK~

Given

H is the midpoint of GK.

Given

GH = KH~

Def. of midpoint

FGH = JKH~

SSS

Solution (flow proof):


Teac

her N

otes

Mark the congruent segments from "Def. of midpoint step" in your diagram

G

F

H

J

K

Slide 183 / 209

In two-column proofs, the statements in the left column are justified by the reasons on the right-side column. As we read down the table, we can see the thought process laid out.

A D

ECB

1 2

Example

Statements Reasons

Note: for SAS, corresponding congruent sides and angles are needed, which we have.

Slide 184 / 209

Example

A

B

C

DStatements Reasons

1. Given, AC bisects BCD

click ___________

click ___________

click ___________

click ___________

A

Slide 185 / 209

Problem

Q R

S

T

click click

click ___________

click ___________click ___________

click ___________

Slide 186 / 209

Problem

D

FG

E

Statements Reasons

click ___________

click ___________

click ___________ click ___________

click ___________

/ 209

Problem

GHJ

F

H is the midpoint of GJ

click

click ___________

click ___________

click ___________

click ___________

click ___________ click ___________

Slide 188 / 209

Statements Reasons

Problem

A

B

C

DT

click ___________

click ___________click ___________

___________

click ___________

click ___________

click ___________

Slide 189 / 209

Statements Reasons

ProblemD C

A B

lines__ __

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

Slide 190 / 209Problem

P

Q R S

TGiven: R is the midpoint of QS, PQR and TSR are right 's, PR = TR~

__

__

click

click ___________

click ___________

click ___________click ___________

click ___________

click ___________

Slide 191 / 209

Statements Reasons

1)

2)

3)

4)

5)

1)

2)

3)

4)

5)

Given: AC = BD, E is the midpoint of AB and CD

~

~Prove: AEC = BED

A

B

DC

E

Problem

Def. of midpoint

E is the midpoint of AB and CD

SSS

AC = BD~

Def. of midpoint

AE = BE~

Given~AEC = BED

CE = DE~Given

Teac

her N

otes


Statements Reasons

1)

2)

3)

4)

5)

1)

2)

3)

4)

5)

Given: AC = BD, E is the midpoint of AB and CD

~

~Prove: AEC = BED

A

B

DC

E

Problem

Def. of midpoint

E is the midpoint of AB and CD

SSS

AC = BD~

Def. of midpoint

AE = BE~

Given~AEC = BED

CE = DE~Given


Teac

her N

otes

Given1)

2)

3)

4)

5)

Statements Reasons

1)

2)

3)

4)

5) SSS

Def. of midpoint

Def. of midpoint

AC = BD~

CE = DE~

AE = BE~

~AEC = BED

E is the midpoint of AB and CD Given

This example can be solved by matching the statements & reasons with their appropriate location.

Ans. is given below.

/ 209


CPCTCCorresponding Parts of Congruent Triangles are Congruent

Slide 193 / 209

CPCTC says that if two or more triangles are congruent by:

SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent.

Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true.

Slide 194 / 209

Process for proving that two segments or angles are congruent

1. Find two triangles in which the two sides or two angles are corresponding parts

2. Prove that the two triangles are congruent (SSS, SAS, ASA, AAS, HL)

3. State that the two parts are congruent, using as the reason: "corresponding parts of congruent triangles are congruent"

Slide 195 / 209

MN

O

E L

111 Which two triangles might you try to prove congruent in order to prove

A

B

C

D

Ans

wer


MN

O

E L


A

B

C

D


Ans

wer B and D

Slide 196 / 209

MN

O

E L


A

B

C

DA

nsw

er

page2svg

(Answer) / 209

MN

O

E L


A

B

C

D


Ans

wer A and C

Slide 197 / 209

MN

O

E L


A

B

C

D1 2

Ans

wer


MN

O

E L


A

B

C

D1 2


Ans

wer B and D

Slide 198 / 209

MN

O

E L


A

B

C

D

Ans

wer


MN

O

E L


A

B

C

D


Ans

wer B and D

Slide 199 / 209

3. Given

4.

5.

6.

Statements Reasons

3. C is the midpoint of AD

4.

5.

6.

Problem

A

B

C D

E

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

/ 209

Problem A

B

C

D

DB bisects ABC ABD = CBD~

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

_____

Slide 201 / 209

Problem AB

C

D E

We are given that BCA = DCE, BC = CD, and B and D are right angles. Since all right angles are congruent, B = D. With the congruent angles and segments, we can conclude that ABC = EDC by ASA. Therefore, BA = DE by CPCTC.

~ ~~

~ ~

click _________________ ________

_______ ________________________

Slide 202 / 209

7. If alt. int. 's =, then lines ||~

Statements Reasons

Problem W X

P

Z Y

click ___________click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________click ___________click ___________

Slide 203 / 209

Triangle Coordinate Proofs


Slide 204 / 209

Coordinate Triangle Proofs

A coordinate proof places a triangle, or any other geometric figure, into a coordinate plane.

A coordinate proof combines: - the geometric postulates, theorems, and properties, and - the Distance Formula and Midpoint Formula.

The only thing that changes from the proofs we have done earlier is you will need to use the Distance and/or Midpoint Formula to calculate side and segment lengths.

Slide 205 / 209

Midpoint Formula TheoremThe midpoint of a segment joining points with coordinates and is the point with coordinates

(x1, y1)(x2, y2)

The Distance FormulaThe distance 'd' between any two points with coordinates and is given by the formula:(x1, y1) (x2, y2)

d =

To refresh your memory:

page2svg

/ 209

ExampleA (0,4)

B (3,0)C (-3,0) Q (0,0)

Statements Reasons

2.

5.4.3.

7.

6.

1. AC = 5 and AB = 5= segments have = measure

1. Given~2.

5.4.3.

7.

6.

Slide 207 / 209

ProblemProve that points: A(4,1), B(5,6), and C(1,3) forms an isosceles right triangle

1. Plot the points2. Use the distance formula to find side lengths3. Does it satisfy the condition for an isosceles

A(4,-1)

B(5,6)

C(1,3)

d =

Distance Formula

Side lengths:

next

Slide 208 / 209

Continued...

triangle

Slide 209 / 209

Problem

A(1,1)

B(4,4)

C(6,2)

d =

Distance Formula

After we plot the points, we can see that they form a triangle.

page186svg

congruent geometry triangles - njctlcontent.njctl.org/courses/math/geometry/congruent...slide 12...

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