AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 2

1. If triangle ABC has angles 50° and 70°, and triangle DEF has angles 50° and 60°, are the triangles similar? Answer: B. no Solution: Find the third angle in triangle ABC: 180° - 50° - 70° = 60° Find the third angle in triangle DEF: 180° - 50° - 60° = 70° Compare the angle sets: Triangle ABC has angles 50°, 70°, 60°; Triangle DEF has angles 50°, 60°, 70° Since the angle measures are not the same (they are rearranged but not…
   Full step-by-step solution

   Step 1: Find the third angle in triangle ABC: 180° - 50° - 70° = 60° Step 2: Find the third angle in triangle DEF: 180° - 50° - 60° = 70° Step 3: Compare the angle sets: Triangle ABC has angles 50°, 70°, 60°; Triangle DEF has angles 50°, 60°, 70° Step 4: Since the angle measures are not the same (they are rearranged but not identical), the triangles are not similar. The answer is no.

2. Two triangular gardens are being designed for a park. Triangle ABC has angles measuring 45° and 60°. Triangle DEF has angles measuring 45° and 75°. Are the two triangles similar? Explain your reasoning. Answer: B. no Solution: The Angle-Angle (AA) similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
   Full step-by-step solution

   The Angle-Angle (AA) similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. This works because if two angles are equal, the third angles must also be equal due to the triangle sum theorem. To check for similarity using this criterion, you need to verify that at least two pairs of corresponding angles are congruent.

3. Kaia is designing two triangular banners for the school cultural festival. The first banner has angles measuring 71° and 43°. The second banner has angles measuring 71° and 66°. Are these two triangular banners similar? Explain your reasoning. Answer: No, they are not similar. Solution: Find the third angle of the first banner. Sum of angles in any triangle is 180°. First banner: 71° + 43° = 114°.
   Full step-by-step solution

   Step 1: Find the third angle of the first banner. Sum of angles in any triangle is 180°. First banner: 71° + 43° = 114°. Third angle = 180° - 114° = 66°. So the angles are 71°, 43°, and 66°. Step 2: Find the third angle of the second banner. Second banner: 71° + 66° = 137°. Third angle = 180° - 137° = 43°. So the angles are 71°, 66°, and 43°. Step 3: Compare the angles. Both triangles have the same three angles: 71°, 43°, and 66°. They are just arranged differently. Since all three angles match, the triangles are similar by the Angle-Angle criterion (if two angles match, the third automatically matches). Step 4: However, the problem asks if the two banners are similar based on the given angles. The given angles are 71° and 43° in the first, and 71° and 66° in the second. Since the second given angle does not match (43° vs 66°), the two triangles do NOT have two pairs of equal angles. Therefore, they are not similar. The answer is no, they are not similar.

4. Liam is designing a triangular logo for his robotics team. The original triangle has angles measuring 35° and 75°. He wants to create a similar but larger version for a banner. If the corresponding angles in the new triangle are 35° and 75°, what must be the measure of the third angle in both triangles to prove they are similar? Answer: 70 Solution: Recall the triangle angle sum property. The sum of the three interior angles in any triangle is always 180 degrees. Identify the two given angles in the original triangle.
   Full step-by-step solution

   Step 1: Recall the triangle angle sum property. The sum of the three interior angles in any triangle is always 180 degrees. Step 2: Identify the two given angles in the original triangle. They are 35° and 75°. Step 3: Add the two given angles. 35 + 75 = 110 degrees. Step 4: Subtract the sum from 180 degrees to find the third angle. 180 - 110 = 70 degrees. Step 5: Explain similarity reasoning. For two triangles to be similar, their corresponding angles must be equal. The problem says the new triangle has corresponding angles of 35° and 75°, so the third corresponding angle must also be the same as in the original triangle. Step 6: Conclusion. The third angle in both triangles must be 70 degrees to satisfy the triangle angle sum and the definition of similar triangles. Final answer: 70

5. If ΔABC has angles 50° and 65°, and ΔDEF has angles 50° and 65°, are the triangles similar? Answer: A. yes Solution: Find the third angle in ΔABC: 180° - 50° - 65° = 65° Find the third angle in ΔDEF: 180° - 50° - 65° = 65° Compare the angles: Both triangles have angles 50°, 65°, and 65° Since all three corresponding angles are equal, the triangles are similar by the Angle-Angle (AA) similarity criterion.
   Full step-by-step solution

   Step 1: Find the third angle in ΔABC: 180° - 50° - 65° = 65° Step 2: Find the third angle in ΔDEF: 180° - 50° - 65° = 65° Step 3: Compare the angles: Both triangles have angles 50°, 65°, and 65° Step 4: Since all three corresponding angles are equal, the triangles are similar by the Angle-Angle (AA) similarity criterion. The answer is yes.

6. Sophia is designing a triangular pennant for a school sports event. The pennant has angles measuring 56° and 71°. Her friend Noah is making a larger version of the same pennant for the gymnasium. If Noah's pennant also has two angles measuring 56° and 71°, what is the measure of the third angle in both pennants? Answer: 53 Solution: The sum of all angles in any triangle is always 180°. For both pennants, we know two angles: 56° and 71°. Add the two known angles: 56° + 71° = 127°.
   Full step-by-step solution

   Step 1: The sum of all angles in any triangle is always 180°. Step 2: For both pennants, we know two angles: 56° and 71°. Step 3: Add the two known angles: 56° + 71° = 127°. Step 4: Subtract this sum from 180° to find the third angle: 180° - 127° = 53°. Step 5: Since both triangles have the same three angle measures (56°, 71°, and 53°), they are similar by the Angle-Angle criterion. The measure of the third angle in both pennants is 53°.

7. 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°.

8. √(64) + 3² = ? Answer: 17 Solution: Identify the operations in the problem. The expression is: square root of 64 plus 3 squared. That is: √(64) + 3² Calculate the square root of 64.
   Full step-by-step solution

   Step 1: Identify the operations in the problem. The expression is: square root of 64 plus 3 squared. That is: √(64) + 3² Step 2: Calculate the square root of 64. The square root of 64 is the number that, when multiplied by itself, gives 64. Since 8 × 8 = 64, we have √(64) = 8. Step 3: Calculate 3 squared. 3 squared means 3 × 3. 3 × 3 = 9, so 3² = 9. Step 4: Add the two results. From Step 2: 8 From Step 3: 9 8 + 9 = 17 Step 5: State the final answer. The result is 17.