Which similarity criterion proves the two triangles are similar in dilation

Question

(1 point)
Responses

AA~
AA~

SSS~
SSS~

SAS~
SAS~

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)

0 / 10000 Word Limit
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Answer 1

When determining if two triangles are similar under dilation, we typically refer to the Angle-Angle criterion, denoted as AA~. According to the AA criterion, if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Answer:

AA~ (Angle-Angle Similarity)

Explanation and Proof:

1. Understanding Dilation: Dilation is a transformation that alters the size of a figure but maintains its shape. This means that the angles of the original triangle remain the same after dilation, while the sides are proportionally scaled.

2. Identifying Angles: Let’s denote triangle ABC and triangle A'B'C' where triangle A'B'C' is a dilation of triangle ABC centered at a point O, with a scale factor k (where k > 0).

   • Triangle ABC has angles ∠A, ∠B, and ∠C.
   • Triangle A'B'C' has corresponding angles ∠A', ∠B', and ∠C'.

3. Proving Angle Equality: Since dilation does not change the angles, it follows that:

   • ∠A = ∠A'
   • ∠B = ∠B'
   • ∠C = ∠C'

4. Applying AA Criterion: To prove that triangles ABC and A'B'C' are similar, we note:

   • We have established that two angles of triangle ABC (∠A and ∠B) are equal to two angles of triangle A'B'C' (∠A' and ∠B').
   • According to the AA similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

5. Conclusion: Therefore, by the AA~ criterion, triangles ABC and A'B'C' are similar.

In conclusion, the similarity criterion that proves two triangles are similar in the context of dilation is AA~ (Angle-Angle Similarity).