Geometry Section 6 Review

Congruence homework from the workbook

Page 75 Item B

Page 76 Example 2 a)

Page 78 Example 4 a) and b)

Page 80 Exercise 2

Page 82 Example 2 a)

Page 83 Example 3 a) and b)

Page 85 Example 6 a) and b)

Page 86 Exercise 2, 6, 8, and 10

Page 88 Example 2

Page 89 Example 4 a) and b)

Page 90 Exercise 2 and 6 to 11

Page 93 Example 3 a) and b)

Page 94 Example 5 a) and b)

Page 96 Exercise 5 and 6

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1. Consider the diagram below of an equilateral triangle.

How long is each side of the triangle? Justify your answer.

ሺ7𝑛ሻ 𝑓𝑡

൬10𝑛 −9

4൰ 𝑓𝑡

൬2𝑛 +15

4൰ 𝑓𝑡

2. Match the description to the type of triangle that is produced.

Description Type of Triangle

a)_______ One Obtuse Angle i. Equilateral

b)_______ All 60° angles ii. Acute

c)_______ No Congruent sides iii. Obtuse

d)_______ One Right Angle iv. Equiangular

e)_______ Three Congruent Sides v. Isosceles

f)_______ Three Acute Angles vi. Scalene

g)_______ Two Congruent Sides vii. Right

3. Consider the figure below.

Part A: Mrs. Konsdorf claims that angle 𝑅 is a right angle. Is Mrs. Konsdorf correct? Explain your

reasoning.

G

R

T

Part B: If 𝑇 is transformed under the rule ሺ𝑥, 𝑦ሻ → ሺ𝑥 − 1, 𝑦 − 2ሻ, then does 𝑇′ form a right angle at

∠𝐺𝑅𝑇′?

4. Consider the figure below.

Determine the measure of each interior angle of △ 𝑀𝐴𝑁 and classify the triangle.

𝑚∠𝐴 =

𝑚∠𝑀 =

𝑚∠𝑁 =

∆𝑀𝐴𝑁 is a(n) ________________________ triangle.

5. Triangle 𝐶𝐴𝑇 has vertices at 𝐶ሺ−6, 0ሻ, 𝐴ሺ4, −2ሻ, and 𝑇ሺ5, 3ሻ.

What type of triangle is 𝐶𝐴𝑇?

A Obtuse

B Isosceles

C Equilateral

D Right

ሺ2𝑥 − 164ሻ°

ሺ𝑥ሻ°

ሺ𝑥 − 48ሻ° N A

M

6. Deloris wants to cover a parallelogram-shaped area of her backyard with yellow,

concrete patio stones. Each stone costs $6.42 and covers 50 square inches. The

parallelogram-shaped area in the backyard has a height of nine feet and a base of

12 feet.

Part A: What is the minimum number of stones that Deloris should buy to cover the

parallelogram-shaped area in her backyard? Show your work below.

Part B: How much money is Deloris going to spend on the yellow, concrete patio?

7. A rectangular banner is 5 inches longer than its width. A triangular poster is three times

as long as its height. Both the poster and the banner have an area of 24 square

inches.

Part A: What is the height and the base of the poster? Justify your answer.

Part B: What is the length and width of the banner? Justify your answer.

8. Consider the parallelogram on the right with an area of 754 𝑐𝑚2.

Part A: Determine the value of 𝑥.

Part B: Determine the height of the parallelogram.

16

9𝑥 − 12

9. Find the area of the trapezoid 𝐻𝑂𝑊𝐿 plotted below. Round your answer to the nearest

hundredth.

10. Consider △ 𝑂𝑃𝐷 in the coordinate system below.

Part A: Find the approximate perimeter of the isosceles triangle △ 𝑂𝑃𝐷. Round your answer

to the nearest hundredth.

Part B: If each block is equal to 25𝑓𝑡2, then determine the area of △ 𝑂𝑃𝐷.

𝐻

𝑂 𝑆

𝑊

𝐿

𝑃 𝑂

𝐷

11. Name two triangles that are congruent by ASA.

12. Name two triangles that are congruent by AAS.

13. Circle the words in the highlighted fields that complete the sentence.

Part A: If two angles | sides and the included angle of one triangle are similar |congruent to two

sides and the included angle of a second triangle, then the two triangles are congruent

by the SSS | SAS | AAS| ASA congruence postulate.

Part B: If at least two | three sides of one triangle are congruent to three sides of a second

triangle, then the two triangles are congruent by the SSS | SAS | AAS| ASA congruence

postulate.

𝐺

𝐴

𝐷

T 𝑆

𝐻

𝐹

𝐶

𝑂

𝐺

𝐴

𝐷

R

𝐵

𝑂

𝐹

𝐶

𝑍

14. Consider quadrilateral 𝑴𝑨𝑹𝑲.

Given: 𝑴𝑨̅̅ ̅̅ ̅ ≅ 𝑹𝑲̅̅̅̅̅ and 𝑴𝑨̅̅ ̅̅ ̅ || 𝑹𝑲̅̅̅̅̅

Prove: ∆𝑴𝑨𝑲 ≅ ∆𝑹𝑲𝑨

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

15.Complete the paragraph proof.

Given: 𝑳𝑬̅̅ ̅̅ bisects ∠𝑳 and 𝑳𝑬̅̅ ̅̅ ⊥ 𝑮𝑼̅̅ ̅̅

Prove: ∆𝑳𝑬𝑮 ≅ ∆𝑳𝑬𝑼

𝑳𝑬̅̅ ̅̅ bisects ∠𝑳 is given. ∠𝑮𝑳𝑬 is congruent to ∠𝑼𝑳𝑬 by the definition of an

__________________ _________________. 𝑳𝑬̅̅ ̅̅ is congruent to 𝑳𝑬̅̅ ̅̅ by the _____________________

property of congruence. 𝑳𝑬̅̅ ̅̅ is perpendicular to 𝑮𝑼̅̅ ̅̅ is given, so ∠𝑮𝑬𝑳 and ∠𝑼𝑬𝑳 are right

angles by the __________________________________________________________. Therefore,

∠𝑮𝑬𝑳 is congruent to ∠𝑳𝑬𝑼 because __________________________ are congruent.

So, ∆𝑳𝑬𝑮 ≅ ∆𝑳𝑬𝑼 by _____________________.

M

K

A

R

L

G

U

E

1

2

3

16.For the AAS Theorem to apply, which side of the triangle must be known?

A the included side

B the longest side

C the shortest side

D a non-included side

17.For the ASA Postulate to apply, which side of the triangle must be known?

A the included side

B the longest side

C the shortest side

D a non-included side

17. Consider the figure of overlapping triangles below.

If it is given that ∠𝑻 ≅ ∠𝑷 and 𝒀𝑫̅̅ ̅̅ ≅ 𝑭𝑫̅̅ ̅̅ , then what is needed to prove that ∆𝒀𝑫𝑻 ≅ ∆𝑭𝑫𝑷

using AAS?

18. Determine which two triangles are congruent by ASA. Justify your answer.

P

Y

T D

F

O T

H S

F

19. Determine if ∆𝑆𝐿𝑃 ≅ ∆𝑍𝐴𝑃 are congruent. Justify your answer.

20. Complete the reasons for the statements below on the two-column proof below. (Hint:

make markings on the triangles)

Given: 𝐻𝑇̅̅ ̅̅ || 𝑆𝐹̅̅̅̅ , ∠𝐻 ≅ ∠𝑆 and 𝐻𝑇̅̅ ̅̅ ≅ 𝑆𝐹̅̅̅̅

Prove: ∆𝐻𝑂𝑇 ≅ ∆𝑆𝑇𝐹

Statements Reasons

1. 𝐻𝑇̅̅ ̅̅ || 𝑆𝐹̅̅̅̅ , ∠𝐻 ≅ ∠𝑆 and 𝐻𝑇̅̅ ̅̅ ≅ 𝑆𝐹̅̅̅̅ 1.

2. ∠𝐹 ≅ ∠𝐻𝑇𝑂 2.

3. ∆𝐻𝑂𝑇 ≅ ∆𝑆𝑇𝐹 3.

L

S

P

A

Z

21. Consider the triangle below.

Determine the angle measure for ∠𝐴 and ∠𝑀 in order for ∆𝐺𝐴𝑀 to be classified as an

isosceles triangle.

22. Determine the measures of ∠𝑁 and ∠𝑂 in ∆𝑁𝑂𝑅 below. List the degree measures from

smallest to largest.

23. Identify the isosceles triangle below along with the base angles.

𝑁

𝑂

𝑅

20′ 72′′

14′ 22.5′

18.6′ 𝐼

𝐻

𝑇

𝐴

64°

𝑀

𝐴

𝐺

24. Complete the two – column proof below with the word bank provided.

Given: Figure GORYAL is equilateral and equiangular.

Prove: ∆𝑌𝐿𝐴 is an isosceles triangle.

A. Equilateral figures have congruent

sides

B. Isosceles Triangle Definition

C. Reflexive Property D. Side – Angle – Side

E. Corresponding Parts of

Corresponding Triangles are

Congruent.

F. An equiangular figure has all angles

that are congruent

G. Transitive Property H. Base Angle Theorem

I. Angle – Angle – Side J. Angle Addition Postulate.

Statement Reasons

1. GORYAL is equilateral and

equiangular.

1. Given

2. 𝑅𝐿 ̅̅ ̅̅ ≅ 𝑂𝐴̅̅ ̅̅ 2.

3. 𝐿𝐴̅̅̅̅ ≅ 𝐿𝐴̅̅̅̅ 3.

4. ∠𝑅𝐿𝐴 ≅ ∠𝑂𝐴𝐿 4.

5. ∆𝑅𝐿𝐴 ≅ ∆𝑂𝐴𝐿 5.

6. ∠𝑌𝐿𝐴 ≅ ∠𝑌𝐴𝐿 6.

7. 𝑌𝐿̅̅̅̅ ≅ 𝑌𝐴̅̅ ̅̅ 7.

8. ∆𝑌𝐿𝐴 8.

𝑂

𝐴

𝐺

𝐿

𝑌

𝑅

25. Consider the diagram to the right.

Determine how ∆𝐺1 can be mapped onto ∆𝐺2.

26. Consider the diagram to the right.

Your best friend determined that ∆𝑀𝑁𝑅 ≅ ∆𝐴𝑃𝑇; however, you overheard someone say

that ∆𝑅𝑁𝑀 ≅ ∆𝑃𝐴𝑇. Determine who is correct and explain why the other answer is

incorrect.

𝐺1

𝐺2

𝑇

𝑀

𝑃

𝐴

𝑁

𝑅

27. Consider the figure below.

Part A: If 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ then because 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ by the

property of congruence, it is possible to determine that ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶

by

Part B: What are the values of 𝑧 and 𝑎?

𝐴

𝐵

𝐶

𝐷

3𝑧 + 2

𝑧 + 6

7𝑎 + 5

5𝑎 + 9

A transitive

B symmetric

C supplement

D reflexive

A AAS.

B ASA.

C SAS.

D SSS.

28. Consider the statement below:

Congruent triangles are always similar.

Which of the following statements is an example of the statement above? Select all that

apply.

Angles are the same, but sides are proportional to each other.

Sides are the same size.

A dilation of a scale factor ≠ 1.

Corresponding angles and corresponding sides are congruent.

A dilation of a scale factor of 1.

29. Are the following triangles similar? Justify your answer.

30. Before rock climbing, Fernando, who’s 5.5 𝑓𝑡. tall, wants to know how high he will

climb. He places a mirror on the ground and walks six feet backwards until he can see

the top of the cliff in the mirror.

Determine the similarity theorem or postulate that you can use to determine the height of

the cliff.

If the mirror is 34 feet from the cliff side, determine the height of the cliff.

𝐷

𝐿

𝑁

𝑂

𝐶 49 21

𝑀

40

12

14 6

31. Determine what similarity postulate or theorem we can use to determine the value of

𝑥, the width of the river.

32. A 1.4 𝑚 tall child is standing next to a flagpole. The child’s shadow is 1.2 𝑚 long. At the

same time, the shadow of the flagpole is 7.5 𝑚 long. How tall is the flagpole?

120 𝑓𝑡

135 𝑓𝑡.

90 𝑓𝑡.

𝑥 𝑓𝑡.