AA Similarity

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Answer Key & Explanations

AA Similarity · Grade 8 · Worksheet 3

1. Emma is an architect designing a triangular stained glass window that needs to be similar to a smaller triangular window in the building's blueprint. The blueprint window has angles measuring 52° and 68°. Emma's full-scale window has two angles measuring 52° and 60°. Are the two triangular windows similar? Explain your reasoning. Answer: B. no Solution: The Angle-Angle similarity criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. This works because if two angles match, the third angles must also be equal since the sum of all angles in any triangle is always 180°.
   Full step-by-step solution

   The Angle-Angle similarity criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. This works because if two angles match, the third angles must also be equal since the sum of all angles in any triangle is always 180°. When applying this criterion, you need to verify that corresponding angles are actually equal between the two triangles.

2. If ΔABC has angles 55° and 65°, and ΔDEF has angles 55° and 60°, are the triangles similar? Answer: B. no Solution: Angle C = 180° - 55° - 65° = 60° So ΔABC has angles: 55°, 65°, 60° Angle F = 180° - 55° - 60° = 65° So ΔDEF has angles: 55°, 60°, 65° Both triangles have angles of 55°, 60°, and 65°, but they are arranged differently.
   Full step-by-step solution

   Step 1: Find the third angle in ΔABC Angle C = 180° - 55° - 65° = 60° So ΔABC has angles: 55°, 65°, 60° Step 2: Find the third angle in ΔDEF Angle F = 180° - 55° - 60° = 65° So ΔDEF has angles: 55°, 60°, 65° Step 3: Compare corresponding angles Both triangles have angles of 55°, 60°, and 65°, but they are arranged differently. Step 4: Apply the Angle-Angle Similarity Criterion For triangles to be similar, all three corresponding angles must be equal. Since the angles are not in the same order (ΔABC: 55°, 65°, 60° vs ΔDEF: 55°, 60°, 65°), the triangles are not similar. The answer is no.

3. 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°

4. Mason is designing two triangular park signs for a community project. One sign has angles measuring 47° and 83°. The other sign has angles measuring 50° and 83°. Are these two triangular signs similar? Explain your reasoning. Answer: No, they are not similar because only one pair of angles (83°) matches, and the other angles (47° and 50°) do not match, so the Angle-Angle criterion is not satisfied. Solution: List the given angles for each triangle. First sign: 47° and 83° Second sign: 50° and 83° Check if any two angles match between the triangles.
   Full step-by-step solution

   Step 1: List the given angles for each triangle. First sign: 47° and 83° Second sign: 50° and 83° Step 2: Check if any two angles match between the triangles. Compare: The first sign has 83°, the second sign has 83° — that's one matching pair. Compare: The first sign has 47°, the second sign has 50° — these do not match. Step 3: Determine if the Angle-Angle criterion is satisfied. The AA criterion requires two pairs of equal angles. Here, only one pair (83°) matches. The other angles (47° and 50°) are different. Step 4: Conclusion. Since only one angle matches, the triangles do NOT satisfy the AA similarity criterion. Therefore, the two triangular signs are not similar. The answer is: No, they are not similar.

5. 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°.

6. If two triangles have angles measuring 45°, 60°, and 75°, are they similar? Answer: A. yes Solution: Two triangles are similar if their corresponding angles are equal. This is known as the Angle-Angle-Angle (AAA) similarity criterion. The first triangle has angles: 45°, 60°, and 75°.
   Full step-by-step solution

   Step 1: Understand the definition of similar triangles. Two triangles are similar if their corresponding angles are equal. This is known as the Angle-Angle-Angle (AAA) similarity criterion. Step 2: Identify the angles of the first triangle. The first triangle has angles: 45°, 60°, and 75°. Step 3: Identify the angles of the second triangle. The second triangle also has angles: 45°, 60°, and 75°. Step 4: Check if the corresponding angles match. Compare the angles: - First triangle: 45°, 60°, 75° - Second triangle: 45°, 60°, 75° We can reorder the angles so they correspond: 45° in first triangle matches 45° in second triangle, 60° in first triangle matches 60° in second triangle, 75° in first triangle matches 75° in second triangle. Step 5: Apply the AAA similarity rule. Since all three angles of one triangle are equal to all three angles of the other triangle, the triangles are similar. Step 6: Final conclusion. Yes, the triangles are similar.

7. Liam is designing a triangular logo for his robotics team. The logo has angles measuring 45° and 75°. His friend Noah is creating a similar but larger version of the logo for a banner. If Noah's logo also has two angles measuring 45° and 75°, what is the measure of the third angle in both logos? Answer: 60 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.
   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. The problem states the two angles are 45° and 75°. Step 3: Add the two given angles. 45 + 75 = 120 degrees. Step 4: Subtract the sum from 180 degrees to find the third angle. 180 - 120 = 60 degrees. Step 5: Explain why both logos have the same third angle. Since both triangles have the same two angle measures (45° and 75°), they are similar triangles. Similar triangles have equal corresponding angles, so the third angle must be the same in both logos. Final answer: The measure of the third angle in both logos is 60 degrees.