1

C'

B'

C

A

A'

B

D

C

A

B

Name: ________________________________________________________________ Date: __________ Day 8: Circumcenter and Incenter Geometry CC Module 1 A Opening Exercise: a) Identify the construction that matches each diagram. Diagram 1 Diagram 2 Diagram 3 Diagram 4

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Term Definition Diagram

Point of Concurrency

Where three or more lines ____________ in one place.

Circumcenter

The intersection of the _______________ _____________ of any triangle. *Center of the circumcircle of the triangle*

Incenter

The intersection of the _______________ _____________ of any triangle. *Center of the incircle of the triangle*

Incredibly, in any triangle the three lines for any of the following are concurrent. Perpendicular Bisectors(circumcenter) Angle Bisectors(incenter) Medians(______________) Altitudes(______________)

D

C

A

B

2

A

B C

Example 1: a) Construct the perpendicular bisectors of the three sides of the triangle below.

b) This point of concurrency is called the ____________________________.

Example 2: a) Use the triangle below to construct the angle bisectors of each angle in the triangle.

b) This point of concurrency is called the ____________________________.

3

Example 3: Use the diagram to identify where each point of concurrency will lie.

a) The incenter will lie on (1) AD (2) AE (3) AF (4) GF

b) The circumcenter will lie on (1) AD (2) AE (3) AF (4) GF

Practice NYTS (Now You Try Some)

1. The diagram below shows the construction of the center of the circle circumscribed about . This construction

represents how to find the intersection of

2. Which geometric principle is used in the construction shown below?

1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle

2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle

3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.

4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle

1) the angle bisectors of 2) the medians to the sides of

3) the altitudes to the sides of 4) the perpendicular bisectors of the sides of

4

Name: ________________________________________________________________ Date: __________ Day 9: Pythagorean Theorem Geometry CC Module 1 A

Opening Exercise: CD is the perpendicular bisector of AB at M . Which pair of segments does not have to be congruent in the construction shown?

1) ,AM BM

2) ,AC BC

3) ,CM DM

4) ,AD BD

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Term Definition Diagram

Pythagorean Theorem

2 2 2a b c

The __________ lengths of any _________ triangle

satisfy the Pythagorean theorem.

Common Pythagorean Triples 3, 4, 5

5, 12, 13 8, 15, 17 7, 24, 25

(multiples of these numbers also satisfy the Pythagorean theorem.)

Example 1: A 10 foot ladder leans against a building, as shown in the diagram below. If the bottom of the ladder is

placed 6 feet from the base of the wall, how many feet up the wall does the brace reach?

5

Example 2: Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as

shown in the diagram below. What is the length of the diagonal, in yards, that Tanya runs?

1) 50 2) 60 3) 70 4) 80

Example 3: Which set of numbers could be the lengths of the sides of a right triangle?

1)

2)

3)

4)

Example 4: Cole placed a ladder against the side of his house. How many feet from the base of a house must a 39-foot

ladder be placed so that the top of the ladder will reach a point on the house 36 feet from the ground?

Example 5: A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of

the pole. How tall is the flagpole?

1) 8 ft 2) 10 ft 3) 12 ft 4) 26 ft

Example 6: Don placed a ladder against the side of his house as shown in the diagram below. To the nearest foot what is

the distance, x, from the foot of the ladder to the base of the house?

6

Practice NYTS(Now You Try Some!) 1. If the length of a rectangular television screen is 20 inches and its height is 15 inches, what is the length of its

diagonal, in inches?

1) 15 2) 13.2 3) 25 4) 35

2. An 18-foot ladder leans against the wall of a building. The base of the ladder is 9 feet from the building on level

ground. How many feet up the wall, to the nearest tenth of a foot, is the top of the ladder?

3. Which set of numbers does not represent the sides of a right triangle?

1)

2)

3)

4)

4. A 10-foot ladder is placed against the side of a building as shown in figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, it is moved 4 feet closer to the base of the building as shown in figure 2. To the nearest foot, how much further up the building does the ladder now reach? Show how you arrived at your answer.

7

Name: ________________________________________________________________ Date: __________ LDay8and9: Pythagorean Theorem WHO DUNNIT Geometry CC Module 1 A

8

9

Name: ________________________________________________________________ Date: __________ Day 10: Mid-Segments in Triangles Geometry CC Module 1 A Opening Exercise:

1. Campsite A and campsite B are located directly opposite each other on the shores of Lake Omega, as shown in the diagram below. The two campsites form a right triangle with Sam’s position, S. The distance from campsite B to Sam’s position is 1,300 yards, and campsite A is 1,700 yards from his position.

What is the distance from campsite A to campsite B, to the nearest yard? 1) 1,095 2) 1,096 3) 2,140 4) 2,141

2. Find the midpoints of sides DE and FE in the triangle below using the perpendicular bisector construction. Label

the points M and N then connect them to create mid-segment MN

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Term Definition Diagram

Mid-Segments in a

The segment joining two ____________ of two sides of a triangle will always be :

______________ to the third side.

_____ the length of the third side.

||DE AC

1

2DE AC

ED

A

B

C

10

Example 1: In the diagram below, joins the midpoints of two sides of . Which statement is not true?

1)

2)

3) 2DC AC 4) 2AB DE

Example 2: Determine the missing information.

A. x = ______ cm y = ______ cm

B. x = ______ cm y = ______ cm

C. x = ______ cm y = ______ cm

D. x = ______ cm

E.

x = ______ cm

F. x = ______ cm y = ______ cm

Example 3: In the diagram below of , is a midsegment of , , , and . Find the

perimeter of .

y

7 cm

x

10 cm

y

8 cm x

22 cm

y

12.5 cmx

7 cm

2x + 3 cm

34 cm

8 cm

8.5 cm

y

x

5x - 4 cm

2x + 1 cm

11

Example 4: Determine the missing information.

x = ______ cm y = ______ cm

x = ______ cm y = ______ cm

x = ______ cm y = ______ cm

Example 5: In the diagram of shown below, D is the midpoint of , E is the midpoint of , and F is the

midpoint of .

If , , and , what is the perimeter of trapezoid ABEF? 1) 24 2) 36 3) 40 4) 44 Practice NYTS (Now You Try Some!)

1.

x = ______ cm y = ______ cm

2.

x = ______ cm

7 cm

6 cm5 cm

y

x

x

6 cm

5 cm

4 cm

y

14 cm

18 cm

19 cm

y

x

8.5 cm

x

y

14 cm

12

C'

B'

C

A

A'

B

D

C

A

B

3.

x = ______ cm

4.

x = ______ cm y = ______ cm

5. As shown in the diagram below, M, R, and T are midpoints of the sides of . If , , and , what is the perimeter of quadrilateral ACRM?

1) 35 2) 32 3) 24 4) 21

6. In the diagram below of , and are midsegments. If , and , determine and state

the perimeter of quadrilateral FDEC.

7. Identify the construction that matches each diagram.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

16 cm6 cm

y

x

5x + 1 cm

13 cm

D

C

A

B

13

Name: ________________________________________________________________ Date: __________ Day 11: Angle Sum of Triangles/Isosceles Triangles Geometry CC Module 1 A

Opening Exercise: In the diagram of shown below, D is the midpoint of , E is the midpoint of , and F is the

midpoint of . If , , and , what is the perimeter of parallelogram ADEF?

Fill in the “Fact/Discovery” column based on geometry facts you have learned!

Fact Diagram

Sum of

The 3 angles of any triangle sum to _______.

Isosceles

A triangle with 2 congruent __________ and 2 congruent ___________ _______________.

Straight Angle

An angle that measures exactly _______ .

Side Lengths in a Triangle

LONGEST SIDE of a triangle is always opposite the __________________ ANGLE.

Angle Measures in a Triangle

SHORTEST SIDE of a triangle is always opposite the __________________ ANGLE.

Ð D

D

14

Example 1: Determine the measure of the missing angles in each triangle below then name the longest side of the

triangle.

a) mA = ________ b) mC = ________ c) mC = ________

b) Longest side:____________ b) Longest side:____________ b) Longest side:____________

Example 2: In the diagram below of isosceles triangle ABC, and angle bisectors , , and are drawn

and intersect at X. If 60om BAC , find .

Example 3: Determine the measure of the numbered angles and EXPLAIN at least one fact you know about triangles that helped you reach your answer.

m1 = _______

m2 = ______

41°

99°

A

B

C

118°

A

B

C

48°

A

B

C

21

27°76°

15

Example 4: In the diagram below of , H is a point on , , , and . Determine whether is an isosceles triangle and EXPLAIN at least two facts you know about triangles that helped you reach your answer.

Practice NYTS (Now You Try Some!) 1.The accompanying diagram shows the roof of a house that is in the shape of an isosceles triangle. The vertex angle

formed at the peak of the roof is 84°. What is the measure of x?

1) 138° 2) 96° 3) 84° 4) 48°

2. Determine the measure of the missing angles in each triangle below then name the longest side of the triangle.

a) x = ________ b) mC = ________ c) mC = ________

b) Longest side:____________ b) Longest side:____________ b) Longest side:____________

3. In the accompanying diagram of , is an equilateral triangle and . What is the value of x, in

degrees? Explain at least 2 facts you know about triangles to show how you reached your solution.

16

Name: ________________________________________________________________ Date: __________ Day 10&11LabLesson: Triangle Inequality Geometry CC Module 1 A Opening Exercise:

For each statement, fill in the circle with >, < or =. For #4-5 use the diagram above

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TRIANGLE INVESTIGATION ACTIVITY!!

TASK #1

You will pick three sides from your bag that create a triangle, and record the measures of each side of the triangle from

shortest to longest; then, find the sum of the measures of the short and medium sides. Repeat this activity twice, with

two other triangles, to complete the chart. [The lengths of the exploragons are labeled along the side of each piece]

TASK #2

You will pick three sides from your bag so that it is impossible to create a triangle, and record the measures of each side

of the non-triangle from shortest to longest; then, find the sum of the measures of the short and medium sides. Repeat

this activity twice, with two other non- triangles, to complete the chart. [The lengths of the exploragons are labeled

along the side of each piece]

Short side Medium side Long side Small + medium

1

2

3

Short side Medium side Long side Short + medium

1

2

3

During this activity, you will compare the sum of the

measures of any two sides of a triangle with the measure

of the third side

17

TASK #3

A) Compare the sum of the measures of the small and medium sides to the measure of the large side for each SUCCESSFUL triangle you created. Fill in the blank to describe what you notice.

TRIANGLE INEQUALITY THEOREM

In order to side lengths to create a triangle the _______ of the 2 smaller sides must be ___________than the longest side.

Practice on Socrative!

Socrative.com or Socrative the app

Student Login - Room Name: WISEYMEPHAM

18

19

Name: ________________________________________________________________ Date: __________ Day 12: Exterior Angle Theorem Geometry CC Module 1 A Opening Exercise: In the diagram below, is isosceles with . If and , what is

? Explain at least 2 facts you know about triangles to show how you reached your solution.

Discovery!

In the diagram of below, , , and is extended through N. Determine the measure of the angles below.

_______

_______

m LKM

m LKN

What do you notice about the measure of LKN compared to the measures of L and M ?

Fact Diagram

Exterior of a Theorem

The measure of an exterior angle of a triangle is equal to the ______ of the measures of the two _______________ interior angles of the triangle.

m ACD m A m B

Ð D

20

Example 1: Find 2m . Example 2: Find 5m , 6m and 7m .

Example 3: Given with 56om B and side extended to D, as shown below. Which value of x makes

?

1) 59º 2) 62º 3) 118º 4) 121º

Example 4: In the diagram below of with side extended through D, 50om A and 120om BCD . Find the missing angles of and determine which side of is the longest side? Justify your answer.

Example 5: In the diagram of below, is extended to point D. If , ,

, what is ?

21

Practice NYTS (Now You Try Some!)

1. Determine the 1m . 2. Determine the 4m .

3. Solve for x in the diagram below.

4. Find the value of x

a) x = ____________ b) x = ____________

5. In the diagram below of isosceles , the measure of vertex angle B is 80°. If extends to point D, find

m BCD ?

x

42°

x

121°

22

Name: ________________________________________________________________ Date: __________ LabLessonDay12and 13: Angle Sum of a Triangle Geometry CC Module 1 A

23

24

Name: __________________________________________________________________ Date: __________ Day 13: Module 1 Constructions and Triangle Basics Test Review Geometry CC Determine whether the following are (T)rue or (F)alse.

1. The incenter is the point of concurrency of the perpendicular bisectors of a triangle. T or F 2. A triangle with side lengths 7, 8, 15 will successfully make a triangle. T or F 3. An median in a triangle is a segment from one vertex and perpendicular to the opposite side. T or F 4. A triangle with side lengths 7, 24, 26 successfully make a right triangle. T or F

5. In the diagram below, joins the midpoints of two sides of . Which statement is not true?

1)

2)

3) 2DC AC 4) 2AB DE

6. If the measures of the angles of a triangle are represented by , , and , the triangle is

1) an isosceles triangle

2) a right triangle

3) an acute triangle

4) an equiangular triangle

7. In the diagram below of isosceles , the measure of vertex angle B is 80°. If extends to point D, what is

?

1) 50

2) 80

3) 100

4) 130

25

8. The end of a dog's leash is attached to the top of a 5-foot-tall fence post, as shown in the diagram below. The dog is 7

feet away from the base of the fence post. How long is the leash, to the nearest tenth of a foot?

1) 4.9

2) 8.6

3) 9.0

4) 12.0

9. CD is the perpendicular bisector of AB at M . Which pair of segments does not have to be congruent in the construction shown?

1) ,AM BM

2) ,AC BC

3) ,CM DM

4) ,AD BD

10. In the diagram below of with side extended, 42om A and the exterior angle at C measures 112o .

Which side of is the longest side? Justify your answer.

11. In shown below, L is the midpoint of , M is the midpoint of , and N is the midpoint of . If , , and , the perimeter of parallelogram BMNL is

26

12. A woman has a ladder that is 26 feet long. If she sets the base of the ladder on level ground 10 feet from the side of

a house, how many feet above the ground will the top of the ladder be when it rests against the house?

13. In the diagram below of isosceles triangle MNO, MN ON and and are bisected by and ,

respectively. Segments MS and intersect at T, and . If , then the measure of angle OTM is

Note: Not drawn to scale

14. Construct an equilateral triangle using the segment shown as one of the three equal sides. Leave all construction

marks.

C

D

27

15. Bisect the angle below.

16. Using a compass and a straightedge, construct the circumcenter of ABC . Label it O.

17. Using a compass and a straightedge, construct the incenter of ABC . Label it O.

A

28

18. In the diagram of below, is extended to point D. If , , ,

what is ?

19. Find the measure of the missing numbered angles. Explain at least 2 facts you know about triangles to show how you reached your solutions.

m1 = _______

m2 = ______

m3 = _______

20. Using a compass and straightedge, construct a perpendicular line(altitude) from vertex J to . [Leave all

construction marks.]

21 3

64°