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1. In the diagram below, $$AM=BM=CM$$ and $$\angle BMC+\angle A = 201^\circ.$$ Find $$\angle B$$ in degrees. 2. In the diagram, $$\triangle BDF$$ and $$\triangle ECF$$ have the same area. If $$DB=2,BA=3,$$ and $$AE=4,$$ find the length of $$\overline{EC}.$$ 3. In triangle $$ABC$$$$AB=AC,$$ $$\angle ABC=72^{\circ},$$ and segment $$\overline{BD}$$ bisects $$\angle ABC$$ with point D on side $$\overline{AC}.$$ If point E is on side $$\overline{BC}$$ such that segment $$\overline{DE}$$ is parallel to side $$\overline{AB}$$ and point F is on side $$\overline{AC}$$ such that segment $$\overline{EF}$$ is parallel to segment $$\overline{BD},$$ how many isosceles triangles are in the figure shown? Feb 11, 2021

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You are only allowed to ask one question at a time.

Feb 11, 2021
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1/   ∠MCA  ≅ ∠A      (These angles are 21º larger than  ∠AMC )

∠BMC + ∠AMC = 180º

∠AMC = 1/3(180 - 42) = 46º

∠BMC = 180 - 46 = 138º

∠B = 1/2(180 - 138) = 21º  Feb 11, 2021
edited by jugoslav  Feb 11, 2021
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You are right! Angle B = 23 degrees

jugoslav

Dragan  Feb 14, 2021