Question

In triangle ABC, points N and M lie on sides AB and AC, respectively.Given AM = AN = BM = NC = BC, find m∠BAC.

Answer 1

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°.