Question

Use the image to answer the 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 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 .

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 △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 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 .

Answer 1

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

Reason: Q lies on MN and P lies on AC, and BP is the transversal, so the angle formed by QB with QM equals the angle formed by BP with PA (corresponding angles).