AoPS - Problem

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Problem

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Tags: geometry, equal angles, parallelogram

parmenides51

17.09.2021 00:37

The point $M$ is chosen inside parallelogram $ABCD$ . Show that $\angle MAB$ is congruent to $\angle MCB$ , if and only if $\angle MBA$ and $\angle MDA$ are congruent.

Math_Is_Fun_101

17.09.2021 03:06

Let $M'$ be the point such that $ABMM'$ and $M'MCD$ are parallelograms. Then, $\angle MAB = \angle MCB \iff \angle AMM' = \angle ADM' \iff$ quadrilateral $AMDM'$ is cyclic $\iff \angle MDA = \angle MM'A = \angle MBA$ , as desired. $\blacksquare$