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Question

Triangle upper A upper B upper C has two line segments inside the triangle. Four points are also plotted, three on the triangle’s sides and one inside of the triangle.

Kayla is attempting to prove MQ=12AP. She has already shown that MN=12AC and MN¯¯¯¯¯¯¯¯¯¯ ∥ AC¯¯¯¯¯¯¯¯ by applying the Triangle Midsegment Theorem. Kayla now wants to prove that △MBQ∼△ABP. She notices that △MBQ and △ABP share the angle ∠MBQ. If Kayla can prove ∠BQM≅∠BPA, she can conclude that △MBQ∼△ABP by the AAA Similarity Theorem. Which of the following is the correct reasoning to prove ∠BQM≅∠BPA?

(1 point)
Responses

Because △MBN∼△ABC and Q is the midpoint of MN¯¯¯¯¯¯¯¯¯¯, ∠BQM≅∠BPA.
Because triangle upper M upper B upper N similar to triangle upper A upper B upper C and upper Q is the midpoint of Modifying above upper M upper N with bar , angle upper B upper Q upper M congruent to angle upper B upper P upper A .

Because △MBQ∼△ABP and corresponding angles of similar triangles are congruent, ∠MBQ≅∠BPA.
Because triangle upper M upper B upper Q similar to triangle upper A upper B upper P and corresponding angles of similar triangles are congruent, angle upper M upper B upper Q congruent to angle upper B upper P upper A .

Because MN¯¯¯¯¯¯¯¯¯¯ ∥ AC¯¯¯¯¯¯¯¯ and corresponding angles of parallel lines are congruent, ∠BQM≅∠BPA.
Because Modifying above upper M upper N with bar parallel to Modifying above upper A upper C with bar and corresponding angles of parallel lines are congruent, angle upper B upper Q upper M congruent to angle upper B upper P upper A .

Because MN¯¯¯¯¯¯¯¯¯¯ ∥ AC¯¯¯¯¯¯¯¯ and alternate exterior angles of parallel lines are congruent, ∠BQM≅∠BPA.
Because Modifying above upper M upper N with bar parallel to Modifying above upper A upper C with bar and alternate exterior angles of parallel lines are congruent, angle upper B upper Q upper M congruent to angle upper B upper P upper A .
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Answer 1

To prove that ∠BQM ≅ ∠BPA, we should consider the information given:

1. It has been established that \( MN \parallel AC \).
2. Based on this parallelism, we can use properties of corresponding angles created by a transversal.

The best reasoning to conclude that ∠BQM ≅ ∠BPA is:

Because MN¯¯¯¯¯¯¯¯¯¯ ∥ AC¯¯¯¯¯¯¯¯ and corresponding angles of parallel lines are congruent, ∠BQM ≅ ∠BPA.

This statement uses the fact that if two lines are parallel and a transversal crosses them, then the corresponding angles are congruent, which applies in this situation as explained.