Question

applying angle relationships of isosceles triangles
what is the measure of ∠egf?

what is the measure of ∠cgf?

Explanation:

Step1: Find $\angle EGF$

Since $EG = FG$, $\triangle EGF$ is isosceles. The base angles of an isosceles triangle are equal. The sum of angles in a triangle is $180^{\circ}$. Let $\angle EGF=\angle GEF = x$. Then $x + x+50^{\circ}=180^{\circ}$.

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Step2: Find $\angle CGF$

$\angle CGF$ and $\angle EGF$ are supplementary (they form a linear - pair). If $\angle EGF = 65^{\circ}$, then $\angle CGF=180^{\circ}-\angle EGF$.

$$ \angle CGF=180^{\circ}-65^{\circ}=115^{\circ} $$

Answer:

The measure of $\angle EGF$ is $65^{\circ}$. The measure of $\angle CGF$ is $115^{\circ}$.