Ellie drew ΔLMN, in which m∠LMN = 90°. She then drew ΔPQR, which was a dilation of ΔLMN by a scale factor of one half from the center of dilation at point M. Which of these can be used to prove ΔLMN ~ ΔPQR by the AA similarity postulate?

 m∠P ≅ m∠L; this can be confirmed by translating point P to point L.
 m∠P ≅ m∠N; this can be confirmed by translating point P to point N.
 segment LM = one halfsegment PQ; this can be confirmed by translating point P to point L.
 segment MN = one halfsegment QR; this can be confirmed by translating point R to point N.

The correct choice for proving that ΔLMN is similar to ΔPQR by the AA (Angle-Angle) similarity postulate is:

**m∠P ≅ m∠L; this can be confirmed by translating point P to point L.**

### Reasoning:
1. **ΔLMN and ΔPQR are right triangles.**
   - Since m∠LMN = 90° and ΔPQR is a dilation of ΔLMN, m∠PQR = 90° as well. Therefore, we know that one pair of angles (the right angles) is congruent.

2. **The dilation from point M.**
   - The dilation involves scaling all the sides of ΔLMN by a factor of one half. Dilation preserves angles, so all corresponding angles of ΔLMN and ΔPQR will be congruent.

3. **AA Similarity Postulate.**
   - To prove similarity by the AA postulate, we need two pairs of congruent angles. We've established that m∠LMN = m∠PQR (both are 90°). 
   - Since ΔPQR is a dilation of ΔLMN, the corresponding angle at vertex L of ΔLMN will be congruent to the corresponding angle at vertex P of ΔPQR. Thus, m∠L ≅ m∠P.

This satisfies the AA similarity postulate, proving that ΔLMN ~ ΔPQR.

Would you like more details or have any questions?

Here are some related questions to consider:
1. What happens to the lengths of corresponding sides in similar triangles?
2. How does the AA similarity postulate compare to other similarity postulates like SAS or SSS?
3. Can two triangles be similar if only one pair of corresponding angles is congruent?
4. How does the concept of dilation affect angles and side lengths in triangles?
5. What are other methods to prove triangles similar besides AA?

**Tip:** Dilation always preserves angles, but scales the lengths of the sides by the scale factor.

Ellie drew ΔLMN, in which m∠LMN = 90°. She then drew ΔPQR, which was a dilation of ΔLMN by a scale factor of one half from the center of dilation at point M. Which of these can be used to prove ΔLMN ~ ΔPQR by the AA similarity postulate? m∠P ≅ m∠L; this can be confirmed by translating point P to point L. m∠P ≅ m∠N; this can be confirmed by translating point P to point N. segment LM = one halfsegment PQ; this can be confirmed by translating point P to point L. segment MN = one halfsegment QR; this can be confirmed by translating point R to point N.

Explore how to prove that ΔLMN is similar to ΔPQR by using the AA similarity postulate and a dilation transformation. This problem involves geometry concepts like triangle similarity, angle congruence, and dilation, making it suitable for students in Grades 8-10.

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