Question

if these two figures are similar, what is the measure of the missing angle?

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is $180^{\circ}$.

Step2: Calculate the missing angle

Let the missing angle be $x$. We have $x + 30^{\circ}+120^{\circ}=180^{\circ}$. Then $x=180^{\circ}-(30^{\circ} + 120^{\circ})$.
$x = 30^{\circ}$. Also, for similar triangles, corresponding angles are equal. The given angles in the first triangle are $30^{\circ},30^{\circ},120^{\circ}$, and the missing angle in the second similar - triangle corresponds to one of the $30^{\circ}$ angles in the first triangle.

Answer:

$30$