AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 2

1. 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: A. 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.

2. Two triangular sections of a park are being designed for landscaping. Triangle PQR has angles measuring 35° at vertex P and 80° at vertex Q. Triangle STU has angles measuring 35° at vertex S and 65° at vertex T. Using the Angle-Angle (AA) similarity criterion, determine if these two triangles are similar. Explain your reasoning. Answer: B. no Solution: Angle R = 180° - (35° + 80°) = 180° - 115° = 65° So triangle PQR has angles: 35°, 80°, 65° Angle U = 180° - (35° + 65°) = 180° - 100° = 80° So triangle STU has angles: 35°, 65°, 80° Both triangles have a 35° angle, but the other angles don't match: Triangle PQR: 35°, 80°, 65° Triangle STU: 35°,…
   Full step-by-step solution

   Step 1: Find the third angle in triangle PQR Angle R = 180° - (35° + 80°) = 180° - 115° = 65° So triangle PQR has angles: 35°, 80°, 65° Step 2: Find the third angle in triangle STU Angle U = 180° - (35° + 65°) = 180° - 100° = 80° So triangle STU has angles: 35°, 65°, 80° Step 3: Compare corresponding angles Both triangles have a 35° angle, but the other angles don't match: Triangle PQR: 35°, 80°, 65° Triangle STU: 35°, 65°, 80° Step 4: Apply the AA similarity criterion For triangles to be similar, we need two pairs of equal corresponding angles. While both triangles have a 35° angle, the other angles are arranged differently. There is no second pair of equal corresponding angles. Step 5: Conclusion The triangles are not similar because they don't satisfy the AA similarity criterion.

3. Liam is examining two triangular sails for a model boat. Triangle KLM has angles measuring 47° at vertex K and 83° at vertex L. Triangle NOP has angles measuring 83° at vertex N and 51° at vertex O. Using the Angle-Angle similarity criterion, determine if the two triangular sails are similar. Explain your reasoning. Answer: Yes, the triangles are similar. Solution: Find the third angle of triangle KLM. Angle M = 180° - (47° + 83°) = 180° - 130° = 50° So triangle KLM has angles: 47°, 83°, 50° Find the third angle of triangle NOP.
   Full step-by-step solution

   Step 1: Find the third angle of triangle KLM. Angle M = 180° - (47° + 83°) = 180° - 130° = 50° So triangle KLM has angles: 47°, 83°, 50° Step 2: Find the third angle of triangle NOP. Angle P = 180° - (83° + 51°) = 180° - 134° = 46° So triangle NOP has angles: 83°, 51°, 46° Step 3: Compare the angle measures of both triangles. Triangle KLM: 47°, 83°, 50° Triangle NOP: 83°, 51°, 46° Step 4: Check for two pairs of equal angles. Both triangles have an 83° angle. The other angles are different: 47° ≠ 51° and 50° ≠ 46°. So only one pair of equal angles exists. Step 5: Apply the AA similarity criterion. The AA criterion requires two pairs of equal angles. Since only one pair matches, the triangles are not similar. The answer is no, the triangles are not similar.

4. If triangle ABC has angles 50° and 65°, and triangle DEF has angles 50° and 65°, are the triangles similar? Answer: B. yes Solution: In triangle ABC, the given angles are 50° and 65°. Calculate the third angle in triangle ABC: 180° - 50° - 65° = 65°. In triangle DEF, the given angles are 50° and 65°.
   Full step-by-step solution

   Step 1: In triangle ABC, the given angles are 50° and 65°. Step 2: Calculate the third angle in triangle ABC: 180° - 50° - 65° = 65°. Step 3: In triangle DEF, the given angles are 50° and 65°. Step 4: Calculate the third angle in triangle DEF: 180° - 50° - 65° = 65°. Step 5: Compare the angles of both triangles: Triangle ABC has angles 50°, 65°, 65°. Triangle DEF has angles 50°, 65°, 65°. Step 6: Since all three corresponding angles are equal, the triangles are similar by the Angle-Angle (AA) criterion. The answer is yes.

5. Liam is designing a triangular logo for his robotics team. He draws triangle ABC with angles measuring 50° and 70°. He wants to create a similar, smaller triangle DEF for a badge. If angle D measures 50° and angle E measures 70°, what must be the measure of angle F to prove the triangles are similar by the Angle-Angle criterion? Answer: 60° Solution: We have two triangles: triangle ABC and triangle DEF. Triangle ABC has angles 50° and 70°. Triangle DEF has angles D = 50° and E = 70°.
   Full step-by-step solution

   Step 1: Understand the problem. We have two triangles: triangle ABC and triangle DEF. Triangle ABC has angles 50° and 70°. Triangle DEF has angles D = 50° and E = 70°. We need to find angle F so that the triangles are similar by the Angle-Angle (AA) similarity criterion. Step 2: Recall the AA similarity rule. Two triangles are similar if two angles of one triangle are equal to two angles of the other triangle. Here, angle A = 50° and angle D = 50° are equal. Angle B = 70° and angle E = 70° are equal. So the third angles must also be equal for the triangles to be similar. Step 3: Find the third angle in triangle ABC. The sum of angles in any triangle is 180°. So angle C = 180° - (50° + 70°) = 180° - 120° = 60°. Step 4: Apply the same to triangle DEF. Angle F must equal angle C for similarity by AA. So angle F = 60°. Step 5: Conclusion. For triangle DEF to be similar to triangle ABC by AA, angle F must be 60°. Final answer: 60°

6. ∛(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

7. √(64) + 3² - 2³ = ? Answer: 9 Solution: √(64) means the positive square root of 64. Since 8 × 8 = 64, we have √(64) = 8. Evaluate the exponent 3² 3² means 3 × 3 = 9.
   Full step-by-step solution

   Let's solve step-by-step. Step 1: Evaluate the square root √(64) means the positive square root of 64. Since 8 × 8 = 64, we have √(64) = 8. Step 2: Evaluate the exponent 3² 3² means 3 × 3 = 9. Step 3: Evaluate the exponent 2³ 2³ means 2 × 2 × 2 = 8. Step 4: Substitute back into the expression Original: √(64) + 3² − 2³ Substitute: 8 + 9 − 8 Step 5: Perform the addition and subtraction from left to right First: 8 + 9 = 17 Then: 17 − 8 = 9 Final answer: 9