AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 1

1. Emma is creating a geometric art piece using two triangular panels. Triangle LMN has angles measuring 55° at vertex L and 65° at vertex M. Triangle XYZ has angles measuring 55° at vertex X and 65° at vertex Y. Using the Angle-Angle (AA) similarity criterion, determine if these two triangles are similar. Explain your reasoning. Answer: Yes Solution: Identify the given angles in triangle LMN: angle L = 55°, angle M = 65°. Step 2: Identify the given angles in triangle XYZ: angle X = 55°, angle Y = 65°.
   Full step-by-step solution

   Step 1: Identify the given angles in triangle LMN: angle L = 55°, angle M = 65°. Step 2: Identify the given angles in triangle XYZ: angle X = 55°, angle Y = 65°. Step 3: Compare the angles: angle L equals angle X (55° = 55°), and angle M equals angle Y (65° = 65°). Step 4: Since two pairs of corresponding angles are equal, the triangles are similar by the AA similarity criterion. The answer is Yes.

2. If ΔABC ~ ΔDEF with ∠A = 35° and ∠B = 80°, find ∠F = ? Answer: 65 Solution: In triangle ABC, we know ∠A = 35° and ∠B = 80° Find ∠C using the triangle angle sum: ∠C = 180° - (35° + 80°) = 180° - 115° = 65° ∠F corresponds to ∠C, so ∠F = ∠C = 65° The answer is 65.
   Full step-by-step solution

   Step 1: In triangle ABC, we know ∠A = 35° and ∠B = 80° Step 2: Find ∠C using the triangle angle sum: ∠C = 180° - (35° + 80°) = 180° - 115° = 65° Step 3: Since ΔABC ~ ΔDEF, corresponding angles are equal Step 4: ∠F corresponds to ∠C, so ∠F = ∠C = 65° The answer is 65.

3. Sophia is designing a triangular window for a treehouse. The window has angles measuring 71° and 46°. Her friend Noah is building a similar triangular window for a fort, but larger. If Noah's window also has two angles measuring 71° and 46°, what is the measure of the third angle in both windows? Answer: 63 Solution: Recall that the sum of all interior angles in any triangle is always 180°. For Sophia's window, we know two angles: 71° and 46°.
   Full step-by-step solution

   Step 1: Recall that the sum of all interior angles in any triangle is always 180°. Step 2: For Sophia's window, we know two angles: 71° and 46°. Step 3: Find the third angle by subtracting the sum of the known angles from 180°: 180° - 71° - 46° = 63°. Step 4: Since Noah's window has the same two angles (71° and 46°), its third angle must also be 63°. Step 5: Both windows have angles measuring 71°, 46°, and 63°. By the Angle-Angle similarity criterion, two pairs of matching angles (71° and 46°) are enough to confirm the triangles are similar. The measure of the third angle in both windows is 63°.

4. Triangle ABC has angles 42° and 77°. Triangle DEF has angles 42° and 61°. Are the triangles similar? Answer: No Solution: Find the third angle in triangle ABC: 180° - 42° - 77° = 61°. So triangle ABC has angles 42°, 77°, and 61°. Find the third angle in triangle DEF: 180° - 42° - 61° = 77°.
   Full step-by-step solution

   Step 1: Find the third angle in triangle ABC: 180° - 42° - 77° = 61°. So triangle ABC has angles 42°, 77°, and 61°. Step 2: Find the third angle in triangle DEF: 180° - 42° - 61° = 77°. So triangle DEF has angles 42°, 61°, and 77°. Step 3: Compare the angles: Triangle ABC has 42°, 77°, 61°. Triangle DEF has 42°, 61°, 77°. The angles are the same set, but they are not in the same order. However, for similarity, corresponding angles must be equal. Here, angle A (42°) matches angle D (42°). But angle B (77°) does not match angle E (61°); it matches angle F (77°). Since the pairs of given angles (42° and 77° in ABC vs 42° and 61° in DEF) do not match, the triangles are not similar by AA. The answer is No.

5. Triangle PQR has angles measuring 3.8 × 10^1° and 7.2 × 10^1°. Triangle STU has angles measuring 3.8 × 10^1° and 8.0 × 10^1°. Are these triangles similar? Answer with 1 for yes or 0 for no. Answer: 0 Solution: Convert the scientific notation angles to standard form: 3.8 × 10^1 = 38°, 7.2 × 10^1 = 72°, 8.0 × 10^1 = 80° Find the third angle in triangle PQR: 180° - 38° - 72° = 70° Find the third angle in triangle STU: 180° - 38° - 80° = 62° Compare the angle sets: Triangle PQR has angles 38°, 72°, 70°…
   Full step-by-step solution

   Step 1: Convert the scientific notation angles to standard form: 3.8 × 10^1 = 38°, 7.2 × 10^1 = 72°, 8.0 × 10^1 = 80° Step 2: Find the third angle in triangle PQR: 180° - 38° - 72° = 70° Step 3: Find the third angle in triangle STU: 180° - 38° - 80° = 62° Step 4: Compare the angle sets: Triangle PQR has angles 38°, 72°, 70° and triangle STU has angles 38°, 80°, 62° Step 5: Check if both triangles have the same three angles: 38° matches, but 72° ≠ 80° and 70° ≠ 62° Step 6: Since the triangles do not have the same three angles, they are not similar. The answer is 0.

6. Two triangular sections of a park are being landscaped. Triangle PQR has angles measuring 35° at vertex P and 80° at vertex Q. Triangle STU has angles measuring 65° at vertex S and 80° at vertex T. Are these two triangles similar? Explain your reasoning using the Angle-Angle similarity criterion. Answer: A. yes Solution: Angle R = 180° - (35° + 80°) = 180° - 115° = 65° So triangle PQR has angles: 35°, 80°, 65° Angle U = 180° - (65° + 80°) = 180° - 145° = 35° So triangle STU has angles: 65°, 80°, 35° Triangle PQR: 35°, 80°, 65° Triangle STU: 65°, 80°, 35° Both triangles have 35° and 80° angles, just in different…
   Full step-by-step solution

   Step 1: Find the missing angle in triangle PQR Angle R = 180° - (35° + 80°) = 180° - 115° = 65° So triangle PQR has angles: 35°, 80°, 65° Step 2: Find the missing angle in triangle STU Angle U = 180° - (65° + 80°) = 180° - 145° = 35° So triangle STU has angles: 65°, 80°, 35° Step 3: Compare the angle measures Triangle PQR: 35°, 80°, 65° Triangle STU: 65°, 80°, 35° Step 4: Check for two pairs of equal angles Both triangles have 35° and 80° angles, just in different positions Step 5: Apply the Angle-Angle similarity criterion Since both triangles have angles measuring 35° and 80°, they are similar by AA similarity. The answer is yes.

7. Emma is designing a triangular sail for her model sailboat. The sail has angles measuring 55° and 65°. Her friend Noah is making a similar but larger sail for his boat. If Noah's sail also has two angles measuring 55° and 65°, what is the measure of the third angle in both sails? Answer: 60 Solution: Find the sum of the two given angles: 55° + 65° = 120° Subtract this sum from 180° to find the third angle: 180° - 120° = 60° Since both triangles have angles measuring 55° and 65°, their third angles must both be 60° The third angle in both sails is 60°.
   Full step-by-step solution

   Step 1: Find the sum of the two given angles: 55° + 65° = 120° Step 2: Subtract this sum from 180° to find the third angle: 180° - 120° = 60° Step 3: Since both triangles have angles measuring 55° and 65°, their third angles must both be 60° The third angle in both sails is 60°.

8. Triangle ABC has angles 55° and 73°, and triangle DEF has angles 55° and 52°. Are the triangles similar? Answer: No Solution: Find the third angle of triangle ABC: 180° - 55° - 73° = 52°. So triangle ABC has angles 55°, 73°, and 52°. Find the third angle of triangle DEF: 180° - 55° - 52° = 73°.
   Full step-by-step solution

   Step 1: Find the third angle of triangle ABC: 180° - 55° - 73° = 52°. So triangle ABC has angles 55°, 73°, and 52°. Step 2: Find the third angle of triangle DEF: 180° - 55° - 52° = 73°. So triangle DEF has angles 55°, 52°, and 73°. Step 3: Compare the given angles: Triangle ABC is given with 55° and 73°. Triangle DEF is given with 55° and 52°. Only one pair of given angles (55° and 55°) is equal. The other given angles (73° and 52°) are not equal. Step 4: Since only one pair of corresponding angles from the given information is equal, the triangles are NOT similar by the AA criterion. The answer is No.