AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 1

1. 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.

2. Emma is designing a triangular sail for her model sailboat. The sail has angles measuring 55° and 65°. Her friend Noah is building 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: Recall that the sum of all angles in any triangle is 180°. For both triangles, we know two angles: 55° and 65°. Calculate the third angle: 180° - 55° - 65° = 60°.
   Full step-by-step solution

   Step 1: Recall that the sum of all angles in any triangle is 180°. Step 2: For both triangles, we know two angles: 55° and 65°. Step 3: Calculate the third angle: 180° - 55° - 65° = 60°. Step 4: Since both triangles have angles measuring 55°, 65°, and 60°, they are similar by the Angle-Angle criterion. The measure of the third angle in both sails is 60°.

3. Liam is designing a triangular logo for his robotics team. The original triangle has angles measuring 45° and 75°. He wants to create a similar, larger version for a banner. If the corresponding angles in the larger triangle are 45° and 75°, and the side between these angles on the original logo is 12 cm, what is the length of the corresponding side on the banner if the scale factor from the original to the banner is 2.5? Answer: 30 cm Solution: We have two similar triangles. The original triangle has angles 45° and 75°, so the third angle is 180° - 45° - 75° = 60°. The larger triangle has the same angles (45°, 75°, 60°), so they are similar.
   Full step-by-step solution

   Step 1: Understand the problem We have two similar triangles. The original triangle has angles 45° and 75°, so the third angle is 180° - 45° - 75° = 60°. The larger triangle has the same angles (45°, 75°, 60°), so they are similar. Step 2: Identify the given side The side between the 45° and 75° angles in the original triangle is 12 cm. In the larger triangle, the side between the 45° and 75° angles corresponds to the same side. Step 3: Apply the scale factor The scale factor from the original to the banner is 2.5. That means every length in the larger triangle is 2.5 times the corresponding length in the original triangle. Step 4: Calculate the corresponding side Corresponding side on banner = original side × scale factor = 12 cm × 2.5 = 30 cm Step 5: Conclusion The length of the corresponding side on the banner is 30 cm.

4. Two triangles are positioned such that triangle ABC has angles measuring 50° and 60° at vertices A and B respectively. Triangle DEF has angles measuring 50° and 70° at vertices D and E respectively. Are the two triangles similar? Explain your reasoning. Answer: No, the triangles are not similar because their corresponding angles are not equal. Triangle ABC has angles 50°, 60°, and 70° (since 180 - 50 - 60 = 70), while triangle DEF has angles 50°, 70°, and 60° (since 180 - 50 - 70 = 60). Although both triangles contain the same three angle measures, they are not arranged in the same order between corresponding vertices, so the Angle-Angle similarity criterion is not satisfied. Solution: Find the missing angle in triangle ABC. We know the sum of angles in any triangle is 180°. Given angles: A = 50°, B = 60°.
   Full step-by-step solution

   Step 1: Find the missing angle in triangle ABC. We know the sum of angles in any triangle is 180°. Given angles: A = 50°, B = 60°. So, angle C = 180° - 50° - 60° = 70°. Thus, triangle ABC has angles: A = 50°, B = 60°, C = 70°. Step 2: Find the missing angle in triangle DEF. Given angles: D = 50°, E = 70°. So, angle F = 180° - 50° - 70° = 60°. Thus, triangle DEF has angles: D = 50°, E = 70°, F = 60°. Step 3: Compare the angles of both triangles. Triangle ABC: 50°, 60°, 70° Triangle DEF: 50°, 70°, 60° Both triangles have the same set of angle measures: 50°, 60°, and 70°. Step 4: Check similarity using the Angle-Angle (AA) similarity criterion. For two triangles to be similar, their corresponding angles must be equal. We must match vertices in the order given: A corresponds to D, B corresponds to E, C corresponds to F. Compare: Angle A (50°) vs angle D (50°) → equal. Angle B (60°) vs angle E (70°) → not equal. Angle C (70°) vs angle F (60°) → not equal. Since the angles at corresponding vertices are not equal, the AA similarity criterion is not satisfied. Step 5: Conclusion Even though both triangles have the same three angles (50°, 60°, 70°), the correspondence given in the problem does not match equal angles. Therefore, the triangles are not similar.

5. ∛(8 × 27) = ? Answer: 6 Solution: We need to find the cube root of the product of 8 and 27. That is: cube root of (8 × 27). 8 × 27 = 216 So the problem becomes: cube root of 216.
   Full step-by-step solution

   Step 1: Understand the problem We need to find the cube root of the product of 8 and 27. That is: cube root of (8 × 27). Step 2: Multiply inside the cube root 8 × 27 = 216 So the problem becomes: cube root of 216. Step 3: Find the cube root of 216 We need a number that, when multiplied by itself three times, gives 216. Let’s check possible numbers: - 5 × 5 × 5 = 125 (too small) - 6 × 6 × 6 = 36 × 6 = 216 (correct) Step 4: Conclusion Since 6 × 6 × 6 = 216, the cube root of 216 is 6. Final answer: 6

6. Two triangular sections of a roof truss are being designed. Triangle ABC has angles measuring 35° at vertex A and 80° at vertex B. Triangle DEF has angles measuring 65° at vertex D and 80° at vertex E. Are these two triangular sections similar? Explain your reasoning using the Angle-Angle similarity criterion. Answer: A. yes Solution: Sum of angles in a triangle = 180° Angle C = 180° - (35° + 80°) = 180° - 115° = 65° Angle F = 180° - (65° + 80°) = 180° - 145° = 35° Triangle ABC has angles: 35°, 80°, 65° Triangle DEF has angles: 65°, 80°, 35° Both triangles have angles measuring 35° and 80° (just in different orders) Since two…
   Full step-by-step solution

   Step 1: Find the missing angle in triangle ABC Sum of angles in a triangle = 180° Angle C = 180° - (35° + 80°) = 180° - 115° = 65° Step 2: Find the missing angle in triangle DEF Angle F = 180° - (65° + 80°) = 180° - 145° = 35° Step 3: Compare the angles of both triangles Triangle ABC has angles: 35°, 80°, 65° Triangle DEF has angles: 65°, 80°, 35° Step 4: Apply the Angle-Angle similarity criterion Both triangles have angles measuring 35° and 80° (just in different orders) Since two angles of triangle ABC equal two angles of triangle DEF, the triangles are similar by the AA similarity criterion. The answer is yes.

7. If triangle ABC has angles 35° and 65°, and triangle DEF has angles 35° and 80°, are the triangles similar? Answer: B. yes Solution: Find the third angle in triangle ABC: 180° - 35° - 65° = 80° Find the third angle in triangle DEF: 180° - 35° - 80° = 65° Compare the angles: Triangle ABC has 35°, 65°, 80° and triangle DEF has 35°, 80°, 65° Since both triangles have angles measuring 35°, 65°, and 80°, their corresponding angles…
   Full step-by-step solution

   Step 1: Find the third angle in triangle ABC: 180° - 35° - 65° = 80° Step 2: Find the third angle in triangle DEF: 180° - 35° - 80° = 65° Step 3: Compare the angles: Triangle ABC has 35°, 65°, 80° and triangle DEF has 35°, 80°, 65° Step 4: Since both triangles have angles measuring 35°, 65°, and 80°, their corresponding angles are equal Step 5: By the Angle-Angle criterion, the triangles are similar The answer is yes.

8. 2x + 5 = 13 Answer: 4 Solution: We are solving the equation: 2x + 5 = 13 Subtract 5 from both sides of the equation. This is done to isolate the term with the variable (2x) on one side. 2x + 5 - 5 = 13 - 5 2x = 8 Divide both sides by 2.
   Full step-by-step solution

   We are solving the equation: 2x + 5 = 13 Step 1: Subtract 5 from both sides of the equation. This is done to isolate the term with the variable (2x) on one side. 2x + 5 - 5 = 13 - 5 2x = 8 Step 2: Divide both sides by 2. This is done to solve for x. 2x / 2 = 8 / 2 x = 4 So the solution is x = 4.