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

Since \( \triangle EGF\) is isosceles (\(EG = FG\)), the base - angles are equal. Let \( \angle EGF=\angle EFG\).
We know that the sum of angles in a triangle is \(180^{\circ}\). Given \( \angle GEF = 50^{\circ}\), then \( \angle EGF+\angle EFG=180^{\circ}-\angle GEF\).
Since \( \angle EGF=\angle EFG\), we have \(2\angle EGF = 180^{\circ}- 50^{\circ}\).

Step2: Solve for \( \angle EGF\)

\(2\angle EGF=130^{\circ}\), so \( \angle EGF = 65^{\circ}\).

Step3: Use the linear - pair relationship

\(\angle CGF\) and \( \angle EGF\) form a linear pair. A linear pair of angles sums to \(180^{\circ}\).
If \( \angle EGF = 65^{\circ}\), then \( \angle CGF=180^{\circ}-\angle EGF\).

Step4: Solve for \( \angle CGF\)

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

Answer:

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