Question

find m∠q. the diagram is not to scale. 124 56 66 103

Explanation:

Step1: Identify vertical - angle

The vertical - angle to the given $77^{\circ}$ angle has the same measure, so it is also $77^{\circ}$.

Step2: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let the unknown angle be $\angle Q$. The exterior angle is $77^{\circ}$ and one of the non - adjacent interior angles is $47^{\circ}$. We use the formula $\text{Exterior Angle}=\angle Q + 47^{\circ}$.
So, $\angle Q=77^{\circ}-47^{\circ}$.

Step3: Calculate the value of $\angle Q$

$77 - 47=30^{\circ}$ seems wrong. Let's assume the $77^{\circ}$ angle and the angle adjacent to $\angle Q$ are supplementary. The angle adjacent to $\angle Q$ is $180 - 77=103^{\circ}$. Since the sum of angles in a triangle formed at the intersection is $180^{\circ}$, and one of the other angles in that triangle is $47^{\circ}$, then $\angle Q = 180-(47 + 30)=103^{\circ}$ (using the fact that the angle adjacent to the $77^{\circ}$ angle forms a triangle with the $47^{\circ}$ angle and $\angle Q$).

Answer:

$103$