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 \( \angle EGF=\angle GEF\). We know that the sum of angles in a triangle is \(180^{\circ}\). Given one angle \( \angle EFG = 50^{\circ}\), then \( \angle EGF+\angle GEF=180^{\circ}-\angle EFG\).

$$ \angle EGF+\angle GEF = 180^{\circ}- 50^{\circ}=130^{\circ} $$

Since \( \angle EGF=\angle GEF\), then \( \angle EGF=\frac{130^{\circ}}{2}=65^{\circ}\)

Step2: Use the linear - pair property

\( \angle CGF\) and \( \angle EGF\) form a linear pair. A linear pair of angles is supplementary (\( \angle CGF+\angle EGF = 180^{\circ}\))

$$ \angle CGF=180^{\circ}-\angle EGF $$

Substitute \( \angle EGF = 65^{\circ}\) into the equation: \( \angle CGF=180^{\circ}-65^{\circ}=115^{\circ}\)

Answer:

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