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Geometry – Unit 5 Review Name: __________________________ Date: _____________ Block: _______ Topics Covered:
• Triangle Sum Theorem • Exterior Angle Theorem • Triangle Inequality Theorem • Hinge Theorem • Ordering Sides/Angles of Triangles • Triangle Congruence (SSS, SAS, ASA, AAS, HL) • Triangle Congruence Proofs • Corresponding Parts Problems
Basic Problems Triangle Sum Theorem:
Determine the measure of each angle in the diagrams below.
Exterior Angle Theorem: Determine the value of the variable in each diagram below.
Triangle Inequality Theorem:
Determine if the side lengths below could create a triangle. 4,6,7 5,2,1
3,4,5 15,17,32
If the following triplets are the lengths of the sides of a triangle, describe the possible values of x in each problem. 4,8,x 5,1,x
3,x,5 x,10,10
B
A
C
28°
(3𝑥 + 4)°
134°
𝑥° 𝑥°
Hinge Theorem:
Determine the relationship of the two quantities separated by a blank space.
Ordering Sides/Angles of Triangles:
Order the sides of triangle ABC from largest to smallest. Given that FG = 12, GH = 6, and HF = 10, order the angles
of triangle FGH from smallest to largest.
Triangle Congruence:
Determine if the triangles shown can be proven to be congruent. If so, write a congruence statement and the accompanying reason. If not, write “Not Congruent” and explain.
A
B C 57°
60°
63°
Congruence Proofs:
Given: 𝑨𝑩 ∥ 𝑫𝑬, 𝑨𝑪 ≅ 𝑪𝑬
Prove: 𝑨𝑩 ≅ 𝑫𝑬
Statements Reasons
1. 𝐴𝐵 ∥ 𝐷𝐸 1. ____________________________________________
2. ∠𝐴𝐶𝐵 ≅ ∠𝐸𝐶𝐷 2. ____________________________________________
3. 𝐴𝐶 ≅ 𝐶𝐸 3. ____________________________________________
4. ∠𝐴 ≅ ∠𝐸 4. ____________________________________________
5. ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶 5. ____________________________________________
6. 𝐴𝐵 ≅ 𝐷𝐸 6. ____________________________________________
Identify the property being illustrated in each statement below. If ∠𝐴 ≅ ∠𝐵 and ∠𝐵 ≅ ∠𝐶, then ∠𝐴 ≅ ∠𝐶.
𝐵𝐶 ≅ 𝐵𝐶
Corresponding Parts:
Identify all of the corresponding parts in each set of congruent triangles with a congruence statement. ∆𝑄𝑅𝑆 ≅ ∆𝑀𝑁𝑂
Application Problems
1. A lookout tower sits on a network of struts and posts. Leslie measured two angles on the tower.
What is 𝑚∠1? _________
2. Determine the range of values of 𝑥 that will create a triangle in the diagrams below. a. b.
3. Use the Hinge Theorem to determine restrictions on the value of 𝑥.
4. Given that ∠𝐻 ≅ ∠𝐿, what additional information would be necessary to prove ∆𝐼𝐽𝐻 ≅ ∆𝐾𝐽𝐿 by ASA? By AAS?
5. In triangle ABC, 𝑚∠𝐴 = 2𝑥 + 4,𝑚∠𝐵 = 3𝑥 − 5, and 𝑚∠𝐶 = 4𝑥 + 1. Order the sides of the triangle from least to greatest.
6. In triangle ABC, the length of 𝐴𝐵 is 3 greater than the length of 𝐵𝐶. 𝐴𝐶 is twice as long as 𝐵𝐶. The perimeter of the triangle is 19. What is the largest angle in triangle ABC? The smallest?
7. Using the information given below, determine if the two triangles are congruent. If so, write a congruence statement and a reason for saying that the triangles are congruent. For Δ𝐴𝐵𝐶, 𝐴 1,5 ,𝐵 4,2 ,𝐶(6,3) For Δ𝐷𝐸𝐹, 𝐷 1,−3 ,𝐸 −4,−1 ,𝐹(−2,0)
Challenge Problems (many answers possible): Develop a problem involving the Triangle Sum Theorem that has a solution of 𝑥 = 5. Can you develop more than one? How about an Exterior Angle Theorem problem that has a solution of 𝑥 = 12? Which of the triangle congruence theorems could be applied to prove that Δ𝐴𝐵𝐶 ≅ Δ𝐸𝐷𝐶 below? Which ones could not? Why?