Question

what is the measure of ∠egf? 65° what is the measure of ∠cgf? ° 50 90 95 115 50°

Explanation:

Step1: Identify Triangle Type

Triangle \( EGF \) has \( FG = FE \) (marked congruent), so it's isosceles with \( \angle EGF = \angle GEF = 65^\circ \) (given \( \angle EGF \) is \( 65^\circ \), and base angles in isosceles triangle are equal).

Step2: Find \( \angle EFG \)

Sum of angles in a triangle: \( \angle EGF + \angle GEF + \angle EFG = 180^\circ \).
Substitute known angles: \( 65^\circ + 65^\circ + \angle EFG = 180^\circ \).
Solve: \( \angle EFG = 180^\circ - 130^\circ = 50^\circ \) (matches the given \( 50^\circ \) for \( \angle EFB \), consistent).

Step3: Calculate \( \angle CGF \)

\( \angle CGF \) and \( \angle EGF \) are supplementary (linear pair, since \( C, G, E \) are colinear? Wait, no—\( C, G, F \)? Wait, \( C, G, \) and the line through \( G \) and \( F \): \( \angle CGF \) and \( \angle EGF \) form a linear pair? Wait, no, \( \angle CGF \) is adjacent to \( \angle EGF \) on a straight line? Wait, actually, \( \angle CGF + \angle EGF = 180^\circ \)? Wait, no—wait, \( \angle EGF = 65^\circ \), so \( \angle CGF = 180^\circ - 65^\circ = 115^\circ \)? Wait, no, wait: Wait, \( \angle EGF = 65^\circ \), and \( \angle CGF \) is supplementary to \( \angle EGF \) because they are on a straight line ( \( C - G - F \)? Wait, no, the line is \( C - G - \) (other direction). Wait, actually, \( \angle CGF \) and \( \angle EGF \) are supplementary (linear pair) because \( C, G, E \) are not colinear—wait, no, the diagram shows \( C, G, \) and the line through \( G \) and \( F \). Wait, \( \angle CGF \) is adjacent to \( \angle EGF \) such that they form a linear pair? Wait, no, let's re-express: \( \angle CGF = 180^\circ - \angle EGF \). Since \( \angle EGF = 65^\circ \), then \( \angle CGF = 180^\circ - 65^\circ = 115^\circ \). Alternatively, using the straight line at \( G \): the sum of \( \angle CGF \) and \( \angle EGF \) is \( 180^\circ \) (linear pair). Thus, \( \angle CGF = 180^\circ - 65^\circ = 115^\circ \).

Answer:

\( 115^\circ \)