Question

∠bmd is a right angle
prove: ∠amb ≅ ∠cmd

the proof is converted to paragraph form.
which parts of the paragraph proof are incorrect?
∠amc is a right angle is given. ∠bmd is a right angle is also given. since ∠amc is a right angle, ∠bmc and ∠cmd are complementary. since ∠bmd is a right angle, ∠amb and ∠bmc are complementary. by the congruent complements theorem, ∠amb ≅ ∠cmd.

Explanation:

1. Analyze the angle relationships: Given \( \angle AMC \) is a right angle, so \( \angle AMB + \angle BMC = 90^\circ \) (complementary angles, by definition of complementary: two angles that add to \( 90^\circ \)). Given \( \angle BMD \) is a right angle, so \( \angle BMC + \angle CMD = 90^\circ \) (complementary angles).
2. Identify the error: In the paragraph proof, it incorrectly states "Since \( \angle AMC \) is a right angle, \( \angle BMC \) and \( \angle CMD \) are complementary". The correct relationship from \( \angle AMC = 90^\circ \) is \( \angle AMB \) and \( \angle BMC \) are complementary, not \( \angle BMC \) and \( \angle CMD \). Also, "Since \( \angle BMD \) is a right angle, \( \angle AMB \) and \( \angle BMC \) are complementary" is incorrect; from \( \angle BMD = 90^\circ \), it should be \( \angle BMC \) and \( \angle CMD \) are complementary.

Answer:

The incorrect parts are:

• "Since \( \angle AMC \) is a right angle, \( \angle BMC \) and \( \angle CMD \) are complementary" (should be \( \angle AMB \) and \( \angle BMC \) are complementary).
• "Since \( \angle BMD \) is a right angle, \( \angle AMB \) and \( \angle BMC \) are complementary" (should be \( \angle BMC \) and \( \angle CMD \) are complementary).