PSAT Math - How to find an angle in a rhombus

The figure referenced is below.

Rhombus

The sides of a rhombus are congruent by definition, so $\overline{RH} \cong \overline{HO}$ , making $\Delta RHO$ isosceles. It is not equilateral, since $m \angle H = 45 ^{\circ }$ , and an equilateral triangle must have three $60 ^{\circ}$ angles.

Also, consecutive angles of a rhombus are supplementary - as they are with all parallelograms - so

$m \angle HRM = 180^{\circ} - m \angle H = 180^{\circ} - 45 ^{\circ} = 135^{\circ}$

A diagonal of a rhombus bisects its angles, so

$m \angle HRO = \frac{1}{2} m \angle HRM = \frac{1}{2} \cdot 135 ^{\circ } = 67\frac{1}{2}^{\circ }$

Similarly, $m \angle HOR = 67\frac{1}{2}^{\circ }$

This makes $\Delta RHO$ acute.

The correct response is that $\Delta RHO$ is acute and isosceles, but not equilateral.

How to find an angle in a rhombus

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