To prove that triangles $\triangle LMN$ and $\triangle PQR$ are similar by the AA (Angle-Angle) similarity postulate, we need to show that they share two pairs of corresponding angles that are congruent.

Given:

1. $m\angle LMN = 90^\circ$.
2. $\triangle PQR$ is a dilation of $\triangle LMN$ with a scale factor of $\frac{1}{2}$ from the center at point $M$.

Since $\triangle PQR$ is a dilation of $\triangle LMN$ with center $M$, each angle in $\triangle PQR$ will be congruent to the corresponding angle in $\triangle LMN$ (since dilation preserves angles). Thus:

• $m\angle LMN = m\angle PQR = 90^\circ$ (both are right angles).
• The other two pairs of corresponding angles will also be congruent because they are preserved under dilation.

Now, let's review the answer choices:

1. $m\angle P \cong m\angle L$; this can be confirmed by translating point $P$ to point $L$.

   • This statement suggests that $m\angle P$ (in $\triangle PQR$) is congruent to $m\angle L$ (in $\triangle LMN$), which can help in proving similarity if $\angle N$ and $\angle Q$ are also shown to be congruent.

2. $m\angle P \cong m\angle N$; this can be confirmed by translating point $P$ to point $N$.

   • This is incorrect because there’s no basis for $m\angle P$ to be congruent to $m\angle N$ based on the dilation. These are not corresponding angles.

3. $\overline{LM} = \frac{1}{2} \overline{PQ};$ this can be confirmed by translating point $P$ to point $L$.

   • While this is true in terms of side lengths, the AA similarity postulate is based on angles, not on the side lengths.

4. $\overline{MN} = \frac{1}{2} \overline{QR};$ this can be confirmed by translating point $R$ to point $N$.

   • Similarly, this is correct in terms of side lengths, but it doesn’t prove similarity using the AA postulate.

Correct Answer

The correct choice is option 1: $m\angle P \cong m\angle L$, which establishes one pair of congruent corresponding angles. Since $m\angle LMN = m\angle PQR = 90^\circ$, we have the two pairs of congruent angles necessary to prove similarity by the AA postulate.

Would you like further details or have any questions?

Here are 5 follow-up questions to expand on this concept:

1. How does the AA similarity postulate relate to the concept of dilation in geometry?
2. Why is it that dilation preserves angles in a figure?
3. How would the similarity change if the dilation scale factor were greater than 1?
4. Can similarity be proven by comparing side lengths instead of angles? Under what conditions?
5. How would this problem differ if the center of dilation were at a different point, not $M$?

Tip: In similarity transformations, remember that while angle measures remain constant, side lengths scale by the dilation factor.