AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 3

1. Triangle ABC has angles 74° and 48°. Triangle DEF has angles 74° and 58°. Are the triangles similar? Answer: No Solution: Find the third angle in triangle ABC. Sum of angles = 180°, so angle C = 180° - 74° - 48° = 58°. Triangle ABC has angles 74°, 48°, and 58°.
   Full step-by-step solution

   Step 1: Find the third angle in triangle ABC. Sum of angles = 180°, so angle C = 180° - 74° - 48° = 58°. Triangle ABC has angles 74°, 48°, and 58°. Step 2: Find the third angle in triangle DEF. Sum of angles = 180°, so angle F = 180° - 74° - 58° = 48°. Triangle DEF has angles 74°, 58°, and 48°. Step 3: Compare the angles. Triangle ABC: 74°, 48°, 58°. Triangle DEF: 74°, 58°, 48°. The angles are the same set, but they are not in the same order. However, for similarity, corresponding angles must be equal. If we match angle A (74°) to angle D (74°), then angle B (48°) must match angle E (58°) — they are not equal. So the triangles are not similar. Step 4: Since two pairs of corresponding angles are not equal, the triangles are not similar by the AA criterion. The answer is No.

2. Isabella is designing two triangular pennants for a school sports event. The first pennant has angles measuring 72° and 37°. The second pennant has angles measuring 72° and 71°. Are the two pennants similar? Explain your reasoning. Answer: No Solution: Find the third angle of the first pennant. Sum of angles in a triangle is 180°. 180° - 72° - 37° = 71°.
   Full step-by-step solution

   Step 1: Find the third angle of the first pennant. Sum of angles in a triangle is 180°. 180° - 72° - 37° = 71°. So the first pennant has angles 72°, 37°, and 71°. Step 2: Find the third angle of the second pennant. 180° - 72° - 71° = 37°. So the second pennant has angles 72°, 71°, and 37°. Step 3: Compare the angle sets. First pennant: 72°, 37°, 71°. Second pennant: 72°, 71°, 37°. They have the same three angle measures, just in different order. Step 4: Since all three angles are equal, the triangles are similar by the Angle-Angle criterion (two pairs of equal angles guarantee the third pair is equal). The answer is yes. The answer is Yes.

3. If ΔABC ~ ΔDEF with ∠A = 38° and ∠B = 67°, find ∠F = ? Answer: 75 Solution: In triangle ABC, ∠A = 38° and ∠B = 67° Find ∠C using the triangle angle sum: 180° - 38° - 67° = 75° ∠F corresponds to ∠C, so ∠F = ∠C = 75° The answer is 75.
   Full step-by-step solution

   Step 1: In triangle ABC, ∠A = 38° and ∠B = 67° Step 2: Find ∠C using the triangle angle sum: 180° - 38° - 67° = 75° Step 3: Since ΔABC ~ ΔDEF, corresponding angles are equal Step 4: ∠F corresponds to ∠C, so ∠F = ∠C = 75° The answer is 75.

4. Hana is examining two triangular garden plots. Triangle LMN has angles measuring 48° at vertex L and 62° at vertex M. Triangle XYZ has angles measuring 48° at vertex X and 70° at vertex Y. Using the Angle-Angle (AA) similarity criterion, determine if these two triangles are similar. Explain your reasoning. Answer: No, the triangles are not similar. Solution: Find the third angle of triangle LMN. Sum of angles = 180° Angle N = 180° - (48° + 62°) = 180° - 110° = 70° So triangle LMN has angles: 48°, 62°, 70°. Find the third angle of triangle XYZ.
   Full step-by-step solution

   Step 1: Find the third angle of triangle LMN. Sum of angles = 180° Angle N = 180° - (48° + 62°) = 180° - 110° = 70° So triangle LMN has angles: 48°, 62°, 70°. Step 2: Find the third angle of triangle XYZ. Sum of angles = 180° Angle Z = 180° - (48° + 70°) = 180° - 118° = 62° So triangle XYZ has angles: 48°, 70°, 62°. Step 3: Compare the angle sets. Triangle LMN: 48°, 62°, 70° Triangle XYZ: 48°, 70°, 62° Step 4: Check for two pairs of equal angles. Both triangles have a 48° angle. The other angles are 62° and 70° in both triangles, but they are paired differently. However, since both triangles contain exactly the same three angles (48°, 62°, and 70°), two angles from one triangle do equal two angles from the other triangle. For example, 48° matches 48°, and 62° matches 62° (or 70° matches 70°). Step 5: Apply the AA similarity criterion. Since two angles of triangle LMN equal two angles of triangle XYZ, the triangles are similar by the AA similarity criterion. The answer is yes, the triangles are similar.

5. 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°.

6. Ava is designing a triangular quilt pattern. One triangular piece has angles measuring 71° and 46°. Her friend Noah is making a similar triangular piece for a different section of the quilt. If Noah's triangle has angles measuring 71° and 46°, what is the measure of the third angle in both triangles? Answer: 63 Solution: Recall that the sum of all angles in any triangle is 180°. For both triangles, we know two angles: 71° and 46°. Calculate the third angle by subtracting the sum of the known angles from 180°: 180° - 71° - 46° = 63°.
   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: 71° and 46°. Step 3: Calculate the third angle by subtracting the sum of the known angles from 180°: 180° - 71° - 46° = 63°. Step 4: Since both triangles have the same two angle measures (71° and 46°), their third angles are also equal (63°), so they are similar by the Angle-Angle criterion. The measure of the third angle in both triangles is 63°.

7. Triangle ABC has angles 54° and 87°. Triangle DEF has angles 54° and 39°. Are the triangles similar? Answer: Yes Solution: Find the third angle in triangle ABC: 180° - 54° - 87° = 39°. So angles are 54°, 87°, 39°. Find the third angle in triangle DEF: 180° - 54° - 39° = 87°.
   Full step-by-step solution

   Step 1: Find the third angle in triangle ABC: 180° - 54° - 87° = 39°. So angles are 54°, 87°, 39°. Step 2: Find the third angle in triangle DEF: 180° - 54° - 39° = 87°. So angles are 54°, 39°, 87°. Step 3: Compare: Both triangles have angles 54°, 87°, and 39°. Two pairs match (54° and 87° in ABC match 54° and 87° in DEF; also 54° and 39° match). Step 4: By the AA similarity criterion, the triangles are similar. The answer is Yes.

8. Liam is designing a triangular garden plot. He knows that one triangle has angles measuring 35° and 75°, and a second triangle has angles measuring 35° and 70°. Are these two triangles similar? Explain your reasoning. Answer: No, the triangles are not similar because their corresponding angles are not equal. The third angle of the first triangle is 70° (180 - 35 - 75), and the third angle of the second triangle is 75° (180 - 35 - 70). Since the angle measures don't match, the Angle-Angle similarity criterion is not satisfied. Solution: Recall the definition of similar triangles. Two triangles are similar if their corresponding angles are equal. The first triangle has angles 35° and 75°.
   Full step-by-step solution

   Step 1: Recall the definition of similar triangles. Two triangles are similar if their corresponding angles are equal. This is known as the Angle-Angle (AA) similarity criterion. Step 2: Find the third angle of the first triangle. The first triangle has angles 35° and 75°. The sum of angles in any triangle is 180°, so: Third angle = 180 - 35 - 75 = 70°. So the first triangle has angles: 35°, 75°, 70°. Step 3: Find the third angle of the second triangle. The second triangle has angles 35° and 70°. Third angle = 180 - 35 - 70 = 75°. So the second triangle has angles: 35°, 70°, 75°. Step 4: Compare the angle sets. First triangle: 35°, 75°, 70° Second triangle: 35°, 70°, 75° Both triangles have the same three angle measures (35°, 70°, 75°), but they are arranged differently in each triangle. For similarity, the corresponding angles must match. Step 5: Check correspondence. If we try to match the 35° angles (since both have a 35° angle), then: - In the first triangle, the other angles are 75° and 70°. - In the second triangle, the other angles are 70° and 75°. So the remaining angles are not equal in the same order. That is, the second angle of the first triangle (75°) does not equal the second angle of the second triangle (70°), and the third angle of the first triangle (70°) does not equal the third angle of the second triangle (75°). Step 6: Conclusion. Since the corresponding angles are not equal, the Angle-Angle similarity criterion is not satisfied. Therefore, the triangles are not similar.