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 ?

Question

GPT-4o mini · Answer

To prove that $\angle BQM \cong \angle BPA$, Kayla can use the fact that the lines $MN$ and $AC$ are parallel, as established with the Triangle Midsegment Theorem. Since $MN \parallel AC$, the following properties of parallel lines and transversals apply: 1. **Corresponding Angles Postulate**: When a transversal crosses parallel lines, the corresponding angles are congruent. Given that $B$ is on $AC$ and $Q$ is on $MN$, and considering line segments $BQ$ and $AP$ act as transversals to the parallel lines $MN$ and $AC$: - $BQ$ serves as the transversal between the parallel lines $MN$ and $AC$, creating corresponding angles. - Specifically, $\angle BQM$ (formed by line $BQ$ with $MN$) and $\angle BPA$ (formed by line $AP$ with $AC$) are corresponding angles. Thus, by the Corresponding Angles Postulate: $$ \angle BQM \cong \angle BPA $$ This congruence allows Kayla to conclude that $\triangle MBQ \sim \triangle ABP$ by the AAA Similarity Theorem, since they have two pairs of congruent angles (the shared angle $\angle MBQ$ and the corresponding angles $\angle BQM$ and $\angle BPA$). Therefore, the correct reasoning to prove $\angle BQM \cong \angle BPA$ is based on the Corresponding Angles Postulate due to the fact that $MN \parallel AC$ and lines $BQ$ and $AP$ act as transversals.