In triangle ABC, points N and M lie on sides AB and AC, respectively.Given AM = AN = BM = NC = BC, find m∠BAC.
1 answer:
Answer:
36°
Step-by-step explanation:
Please refer to the attached figure.
Triangles ANC, AMB, BCN, CBM are all isosceles by virtue of the given congruences. By symmetry, we also claim that ΔABC is isosceles (AB≅AC).
We aren't sure how their internal angles relate, but we can write a couple of equations.
Let x = m∠BAC, y = m∠BCN.
∠NCA = ∠NAC = ∠MBA = ∠MAB = x
∠BCM = ∠BMC = x +y
∠BMC = ∠MBA +∠MAB = 2x
So, we have ...
∠BMC = x +y = 2x ⇒ y = x
And the sum of angles around ΔABC is ...
x +(2x) +(2x) = 180°
x = 180°/5 = 36°
The measure of angle BAC is 36°.
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