(1)①填空:当∠AMN =90°时,AM=MN; ②证明:取的AB中点P,连结PM ∵四边形ABCD是正方形 ∴∠PAM +∠AMB =90° ∵∠AMN =90° ∴∠CMN+∠AMB =90° ∴∠PAM = CMN ∵点M是边BC的中点 点P是边AB的中点 AB=BC ∴AP=MC BP=BM ∵∠B =90° ∴△BPM是等腰直角三角形 ∴∠BPM =45° ∴∠APM =135° ∵∠DCB =90° ∴∠DCQ =90° ∴∠NCQ =45° ∴∠MCN =135° ∴∠APM =∠MCN ∴△APM ≌△MCQ ∴AM=MN. | |