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 △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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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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Answers
GPT-4o mini
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.
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.