SAT Prep Triangle Equation

Question

I got this problem from my teacher and don't know how to solve, could someone please help?

(SAT Prep) In △ABC, ∠B = 90°, BH = AH, and the ratio of m∠A to m∠C is 1:2. Find m∠BHA?

    A. 100°
    B. 90°
    C. 120°
    D. 60°

I know that m∠A = 30°, and that m∠C = 60°

Answer 1

$\triangle BAH$ is isosceles because it has two equal sides.

This means that $m(\angle BAH) = m\angle ABH$ because of base angle theorem.

Then, $m\angle BHA = 180 - (m(\angle BAH) + m(\angle ABH))$ (because the sum of the measures of the angles in a triangle is $180$)

But since $m(\angle BAH) = m(\angle ABH) = 30$, we simply have

$m(\angle BHA) = 180 - (30 + 30) = 120$

Answer 2

Hint: Let us denote $$\angle{AHB}=\beta$$ and$$\angle{ACH}=\gamma$$. Since we have $AH=HB$ we get $$\beta=\pi-2\alpha$$ and $$\alpha+\gamma=\frac{\pi}{2}$$ since we have $$2\alpha=\gamma$$ we get $$\alpha=30^{\circ}$$ and $$\beta=120^{\circ}$$