Why is angle BMC 90?

[spare bone]

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[finite goblet]

BM and CM are angle bisectors of B and C (as AB=AM and DC=DM).

[finite goblet]

Or even simpler. Just take N midpoint of BC. Then BN=CN=MN. So, CMB=90

[spare bone]

i understand why your 2nd example but why is BM and CM angle bisectors prove that BMC = 90?

[old grotto]

Because the angles ABC and BCD add up to 180° and the bisectors half both of them.