Question

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

Explanation:

Step1: Use the property of isosceles triangle

In \(\triangle EGF\), since \(EG = FG\), \(\angle EGF=\angle GEF\). And we know that the sum of angles in a triangle is \(180^{\circ}\). Let \(\angle EGF = x\), then \(x + x+50^{\circ}=180^{\circ}\).

Step2: Solve the equation for \(\angle EGF\)

$$
LATEXBLOCK0
$$

Step3: Use the linear - pair relationship for \(\angle CGF\)

\(\angle CGF\) and \(\angle EGF\) form a linear pair. So \(\angle CGF=180^{\circ}-\angle EGF\).

Step4: Calculate \(\angle CGF\)

Since \(\angle EGF = 65^{\circ}\), then \(\angle CGF=180^{\circ}-65^{\circ}=115^{\circ}\)

Answer:

\(\angle EGF = 65^{\circ}\), \(\angle CGF=115^{\circ}\)