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www.MathNation.com Section 6 Topic 1

Name ____________________________________________ Date_______________ Triangles – Part 1 Introduction to Triangles – Part 1 Independent Practice 1. Consider the diagram below of an equilateral triangle.

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

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

(7𝑛) 𝑓𝑡

+10𝑛 −940 𝑓𝑡

+2𝑛 +154 0 𝑓𝑡

Independent Practice Packet 1


3. Consider the figure below.

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

Part B: If 𝑇 is transformed under the rule (𝑥, 𝑦) → (𝑥 − 1, 𝑦 − 2), then does 𝑇′ form a right angle

at ∠𝐺𝑅𝑇′?

G

R

T



8

12" #

$ % &

8 3�

4. Consider the triangle below.

Part A: If Δ𝐴𝑀𝐺 is an isosceles triangle with base 𝐴𝐺, what is the value of 𝑥? Justify your

answer. Part B: What is the length of each leg? Part C: What is the length of the base?

5. Consider the diagram on the right. Classify each triangle as equilateral, isosceles, or

scalene.

∆𝐼𝐻𝐺: __________________________

∆𝐻𝐽𝐼: __________________________

∆𝐾𝐻𝐼: __________________________

∆𝐻𝐽𝐾: __________________________

M

G A (3𝑥 –0.4) 𝑖𝑛



Name ____________________________________________ Date_______________ Triangles – Part 1 Introduction to Triangles – Part 2 Independent Practice 1. 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.

2. Triangle 𝐶𝐴𝑇 has vertices at 𝐶(−6, 0), 𝐴(4,−2), and 𝑇(5, 3).

What type of triangle is 𝐶𝐴𝑇?

A Obtuse B Isosceles C Equilateral D Right

(2𝑥 − 164)°

(𝑥)°

(𝑥 − 48)° N A

M



𝑤°

67°

Y

I

K

3. Garden Plus LLC. Is fencing a triangular garden for Mr. Gold (pictured on the right).

Part A: Determine the expression for the measure of angle 𝑌. Part B: If 𝑚∠𝐼 = 33 and 𝑚∠𝑌 = 14𝑑 − 19, then determine the

value of 𝑑.

4. Triangle 𝑂𝑀𝐺 has vertices at 𝑂(4,−2), 𝑀(5, 3), and 𝐺(– 6, 0). If point 𝐺 is transformed under

the translation of 𝑥, 𝑦 → (𝑥 + 3, 𝑦 + 2), then △ 𝑂𝑀𝐺′ is

o equilateral.

o isosceles.

o scalene.



5. Consider the following figure.

Given: ∆𝐴𝐵𝐶, and 𝐵𝑃 is parallel to 𝐴𝐶. Prove: 𝑚∠1 +𝑚∠2 +𝑚∠3 = 180°

Statements Reasons 1. 𝐴𝐵𝐶 is a triangle. 1.

2. 𝐵𝑃 || 𝐴𝐶 2.

3. 𝑚∠1 +𝑚∠5 = 𝑚∠𝑃𝐵𝐴 3.

4. 𝑚∠𝑃𝐵𝐴 +𝑚∠4 = 180 4.

5. 𝑚∠1 +𝑚∠5 +𝑚∠4 = 180° 5.

6. ∠2 ≅ ∠4; ∠3 ≅ ∠5 6.

7. 𝑚∠2 = 𝑚∠4; 𝑚∠3 = 𝑚∠5 7.

8. 𝑚∠1 +𝑚∠2 +𝑚∠3 = 180° 8.

A

B

C

P 4 5 1

2 3



Name ____________________________________________ Date_______________ Triangles – Part 1 Area and Perimeter in the Coordinate Plane – Part 1 Independent Practice 1. 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?

2. 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.



3. The perimeter of a school gym is 522 feet. The gym measures 80 feet wide. Determine the

length of the school gym. A 442 feet B 181 feet C 362 feet D 90.5 feet

4. The massive rectangular map of the world at Switchboard INC. is made of six interlocking

isosceles triangle pieces. Each triangle has a base of five feet and a height of eight feet. If the CEO of Switchboard INC. wants to cover the map in a protective residue with two coats of Mod Podge, then what is the total area of covering for the protective residue?

5. A jumbo playing card has an area of 64 square inches and perimeter of 32 inches. What

are the dimensions of the playing card? _____________ inches by _______________ inches

6. Consider the parallelogram on the right with an area of 754 𝑐𝑚0.

Part A: Determine the value of 𝑥. Part B: Determine the height of the parallelogram.

16

9𝑥 − 12



Name ____________________________________________ Date_______________ Triangles – Part 1 Area and Perimeter in the Coordinate Plane – Part 2 Independent Practice 1. Find the area of the trapezoid 𝐻𝑂𝑊𝐿 plotted below. Round your answer to the nearest

hundredth.

2. Triangle 𝑆𝐵𝐴 has coordinates 𝑆(15,−8), 𝐵(−2,21), and 𝐴(0,0). If the height of the triangle

for the corresponding base 𝑆𝐵 is 8.89 units, then determine the perimeter and area of △𝑆𝐵𝐴. Round your answer to the nearest unit.

𝐻

𝑂 𝑆

𝑊

𝐿



3. 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 25𝑓𝑡8, then determine the area of △ 𝑂𝑃𝐷.

4. Consider the right triangle below.

If the perimeter is 1013 units, find the value of 𝑥 and the area of the triangle.

The value of 𝑥 is ______________. The area is ________________ square units.

𝑃 𝑂

𝐷

16𝑥 + 27

14𝑥 − 45 25𝑥



Name ____________________________________________ Date_______________ Triangles – Part 1 Triangle Congruence – SSS and SAS – Part 1 Independent Practice 1. If ∆𝐺𝑇𝑅 ≅ ∆𝑁𝐴𝑂, then finish the following congruence statements and mark the

corresponding congruent sides and the corresponding congruent angles.

∠𝐴 ≅ ______ 𝑁𝑂 ≅ ______

______ ≅ ∠𝑅 ______≅ 𝐺𝑇

∠𝐺 ≅ ______ 𝑅𝑇 ≅ ______

2. Name two triangles that are congruent by ASA.

3. Name two triangles that are congruent by AAS.

𝐺

𝐴

𝐷

T 𝑆

𝐻

𝐹

𝐶

𝑂

𝐺 𝐴

𝐷

R

𝐵

𝑂

𝐹

𝐶

𝑍

G A

R

T

O N



4. Complete the congruence statements for the triangles below.

∆𝐺𝑅𝐿 ≅ ∆_______ ∠𝑁 ≅ ∠______ 𝐺𝐿 ≅ ______ ∠𝑅 + ∠𝑁 ≅ ∠______ + ∠________

5. 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.

6. Consider the figure of the kite below.

Part A: What information is needed to prove that the triangles above are congruent using

the SSS Congruence Postulate?

Part B: What information is needed to prove that the triangles above are congruent using the SAS Congruence Postulate?

M

K A

R

M

N A L

G

R



Name ____________________________________________ Date_______________ Triangles – Part 1 Triangle Congruence – SSS and SAS – Part 2 Independent Practice 1. Ernie draws ∆𝑀𝐴𝑅 and ∆𝑁𝐼𝐿 where 𝑀𝑅 ≅ 𝑁𝐿, 𝑀𝐴 ≅ 𝑁𝐼, and ∠𝐴 ≅ ∠𝐼. Draw a sketch of

∆𝑀𝐴𝑅 and ∆𝑁𝐼𝐿 to determine if Ernie can use either SSS or SAS to prove the two triangles congruence. If the answer is no, explain what additional information the Ernie needs.

2. Consider quadrilateral 𝑀𝐴𝑅𝐾.

Given: 𝑀𝐴 ≅ 𝑅𝐾 and 𝑀𝐴 || 𝑅𝐾 Prove: ∆𝑀𝐴𝐾 ≅ ∆𝑅𝐾𝐴

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

M

K

A

R



3. Rose claims that since ∆𝑀𝑇𝑊 ≅ ∆𝑅𝐹𝑆 are both equiangular triangles, then they must be congruent by the SSS Congruence Postulate. Determine whether Rose correct or incorrect? Justify your answer.

4. Complete the paragraph proof.

Given: 𝐺𝐼 ≅ 𝐼𝑅 ≅ 𝐺𝐿 ≅ 𝑅𝐿 Prove: ∆𝐺𝐼𝑅 ≅ ∆𝐺𝐿𝑅

𝐺𝑅 is congruent to 𝐺𝑅 by the _____________________ property of congruence. Since it is

given that ___________________________________________, then it is possible to say

∆𝐺𝐼𝑅 ≅ ∆𝐺𝐿𝑅 by _____________________ Congruence Postulate.

5. Draw ∆𝑇𝐻𝑆 and complete the sentences below.

Part A: The angle that is included between 𝐻𝑇 and 𝑆𝑇 is _________________. Part B: ______________ and _______________ include ∠𝑆.

I

L

G R



Name ____________________________________________ Date_______________ Triangles – Part 1 Triangle Congruence – ASA and AAS – Part 1 Independent Practice 1. Complete the paragraph proof.

Given: 𝐿𝐸 bisects ∠𝐿 and 𝐿𝐸 ⊥ 𝐺𝑈 Prove: ∆𝐿𝐸𝐺 ≅ ∆𝐿𝐸𝑈

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

__________________ _________________. 𝐿𝐸 is congruent to 𝐿𝐸 by the _____________________

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

angles by the __________________________________________________________. Therefore,

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

So, ∆𝐿𝐸𝐺 ≅ ∆𝐿𝐸𝑈 by _____________________.

2. 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

3. 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

L

G

U

E



4. Complete the two-column proof by filling the blanks.

Given: ∠𝐽 ≅ ∠𝐻 and line 𝑀 bisects 𝐺𝐾at 𝐿 Prove: ∆𝐽𝐿𝐾 ≅ ∆𝐻𝐿𝐺

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

5. Consider the figure of overlapping triangles below.

If it is given that ∠𝑇 ≅ ∠𝑃 and 𝑌𝐷 ≅ 𝐹𝐷, then what is needed to prove that ∆𝑌𝐷𝑇 ≅ ∆𝐹𝐷𝑃 using AAS?

P

Y

T D

F

M

K

J G

H

L



E A

T

G

R

Name ____________________________________________ Date_______________ Triangles – Part 1 Triangle Congruence – ASA and AAS – Part 2 Independent Practice 1. Complete the two-column proof by filling in statements for the given reasons.

Given: ∠𝑅 and ∠𝐴 are right angles.

𝑅𝐴 bisects 𝐺𝑇. Prove: △ 𝐺𝐸𝑅 ≅△ 𝑇𝐸𝐴

Statements Reasons

1. 1. Given

2. 2. All right angles are congruent

3. 3. Vertical angles are congruent

4. 4. Given

5. 5. Definition of bisector

6. 6. AAS congruence theorem

2. Consider triangle ∆𝑁𝑌𝑀.

Part A: Which side is included between ∠𝑁 and ∠𝑀?

Part B: 𝑌𝑀 is between which angles?



L

I

W

R

M

H

1

2

3

3. Consider the statement, ∆𝑅𝐼𝑀 ≅ ∆𝑊𝐻𝐿 for the figure on the right.

Which of the following statements represents the supports the statement above? A Yes, the triangles are congruent by ASA. B No 𝑅𝐼 and 𝐻𝐿 are not corresponding sides. C Yes, the triangles are congruent by AAS. D No, ∠𝐵 and ∠𝑈 are not corresponding angles.

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

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

L

S

P

A

Z



O T

H S

F

6. Complete the reasons for the statements below on the two-column proof below. (Hint: make markings on the triangles) Given: 𝐻𝑇 || 𝑆𝐹, ∠𝐻 ≅ ∠𝑆 and 𝐻𝑇 ≅ 𝑆𝐹 Prove: ∆𝐻𝑂𝑇 ≅ ∆𝑆𝑇𝐹

Statements Reasons

1. 𝐻𝑇 || 𝑆𝐹, ∠𝐻 ≅ ∠𝑆 and 𝐻𝑇 ≅ 𝑆𝐹 1.

2. ∠𝐹 ≅ ∠𝐻𝑇𝑂 2.

3. ∆𝐻𝑂𝑇 ≅ ∆𝑆𝑇𝐹 3.



Name ____________________________________________ Date_______________ Triangles – Part 1 Using Triangle Congruency to Find Missing Variables Independent Practice 1. Consider the figures below.

Find the value of 𝑥 and 𝑦 in order to prove that the two triangles are congruent by the SAS Congruence Postulate. Justify your work.

2. Consider the figures below.

Find the value of 𝑥 and 𝑦 in order to prove that the two triangles are congruent using the ASA Congruence Postulate. Justify your work.

3𝑦 –712

52°

2456

(14𝑥 − 9)°

(9𝑦 + 7)°

34 𝑥 −

58

2324 𝑥 − 5




Find the values of 𝑘 and 𝑗 that prove the two triangles are congruent using the SSS Congruence Postulate.


Find the values of 𝑥 and 𝑦 that prove the two triangles are congruent using the AAS Congruence Theorem. Justify your work.

𝑘 + 56

5678

10𝑗 − 17

169 𝑘 − 231

:56 𝑦; ° <1692 =°

34𝑥 −421

942




Part A: If 𝐴𝐵 ≅ 𝐴𝐷 and 𝐵𝐶 ≅ 𝐷𝐶 then because 𝐴𝐶 ≅ 𝐴𝐶 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.


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