Which similarity criterion proves the two triangles above are similar?

Question

(1 point)
Responses

SSS~
SSS~

SAS~
SAS~

AA~
AA~

AAS~
AAS~
Question 2
Explain your reasoning by proving the answer above and show your work. You may type it out or upload a picture. You must fully explain and show all work in order to receive full credit. (3 points)

Answer 1

To determine which similarity criterion proves that two triangles are similar, we need to review the criteria for similarity:

1. Angle-Angle Similarity (AA): If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
2. Side-Side-Side (SSS): If the lengths of the corresponding sides of two triangles are in proportion, then the triangles are similar.
3. Side-Angle-Side (SAS): If two sides of one triangle are in proportion to two sides of another triangle and the included angle between those two sides is equal, then the triangles are similar.
4. Angle-Angle-Side (AAS): If two angles of one triangle and a non-included side are equal to two angles and the corresponding non-included side of another triangle, then the triangles are similar. However, this is not often listed as a primary criterion since it can be derived from AA.

To solve the problem, let's assume that we have two triangles, Triangle 1 and Triangle 2, and we are given certain angle measurements and/or side lengths. Depending on the information provided, we would use one of the criteria to prove similarity.

Example Explanation Using AA Criterion:

1. Given Angle Measurements:

   • Triangle 1: ∠A = 30°, ∠B = 60°
   • Triangle 2: ∠X = 30°, ∠Y = 60°

   Here, we can see that two angles in Triangle 1 are equal to two angles in Triangle 2.

2. Applying AA Criterion: Since ∠A = ∠X and ∠B = ∠Y, by the Angle-Angle Similarity Criterion (AA), we can conclude that Triangle 1 is similar to Triangle 2.

3. Conclusion: Therefore, the correct similarity criterion that proves the two triangles are similar is AA~.

Full Explanation of Work (3 points):

1. Identify Known Angles: Based on the information provided, identify the angles in both triangles.
2. Check for Equality: Verify that at least two angles from one triangle match with two angles from the other triangle.
3. State the Criterion Used: Clearly state that AA criterion was used because two pairs of angles are equal.
4. Make Conclusion: Conclude that the triangles are similar based on the AA criterion.

Note:

If you were provided information about side lengths instead, then you would have followed the SSS or SAS criteria as necessary. Please provide the actual measurements or angles if you want to approach a specific triangle similarity case.