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.

Here, m ∠P ≅ m ∠P this can be confirmed by translating point P to point L, can be used to prove ΔLMN ~ ΔPQR by the AA similarity postulate.

Given that,

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.

We have to determine,

Which of these can be used to prove ΔLMN ~ ΔPQR by the AA similarity postulate.

According to the question,

Angle angle (AA) similarity postulate state that two triangles are similar if two of their corresponding angle is similar. The corresponding angle for each point of the triangles will be:

∠L=∠P

∠Q=∠M

∠N=∠R

Triangles are always similar but similar triangles need not be congruent. Two geometrical figures having exactly the same shape and size are said to be congruent figures.

Since the 2nd triangle is made from dilation, it should maintain its orientation.

∠P corresponds to ∠L. If you move/translate point P to point L, you can confirm it because their orientation is the same.

The triangle will be similar if ∠P=∠N but you can't confirm it with translation alone.

Hence,  m ∠P ≅ m ∠P this can be confirmed by translating point P to point L can be used to prove ΔLMN ~ ΔPQR by the AA similarity postulate.

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

Step-by-step explanation:

Angle angle (AA) similarity postulate state that two triangles are similar if two of their corresponding angle is similar. The corresponding angle for each point of the triangles will be:

∠L=∠P

∠Q=∠M

∠N=∠R

Since the 2nd triangle made from dilation, it should maintain its orientation.

Option 1 is true, ∠P corresponds to ∠L. If you move/translate point P to point L, you can confirm it because their orientation is the same.

Option 2 is false, the triangle will be similar if ∠P=∠N but you can't confirm it with translation alone.

Option 3 and 4 definitely wrong because it speaking about length, not the angle.

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