Question

what is the measure of ∠egf?
°
what is the measure of ∠cgf?
40 °
55
65
80
(image shows a geometric figure with points a, e, g, f, b, c. lines and angles: ∠ at f is 50°, some sides marked equal.)

Explanation:

Step1: Identify triangle type

Triangle \( EGF \) has \( EG = FG \) (marked with equal segments), so it's isosceles with \( \angle GEF=\angle GFE \)? Wait, no, \( \angle GFE = 50^\circ \)? Wait, \( FB \) is a straight line, so \( \angle GFB \) is a straight angle, but \( \angle EFG = 50^\circ \)? Wait, actually, \( EG = EF \)? Wait, the marks: \( EG \) and \( EF \)? Wait, the diagram has \( EG \) and \( FG \) with marks? Wait, no, the two sides \( EG \) and \( EF \)? Wait, no, looking at the diagram, \( GF \) and \( EF \) have marks? Wait, the red marks: \( \angle EGF \) and \( \angle GEF \)? Wait, no, let's re-examine. The triangle \( EGF \): sides \( GF \) and \( EF \) are marked equal (the tick marks), so \( \triangle EGF \) is isosceles with \( GF = EF \). Wait, no, the tick marks on \( GF \) and \( EG \)? Wait, the horizontal side \( GF \) has a tick, and the side \( EF \) has a tick? Wait, maybe \( EG = EF \)? Wait, no, the angle at \( F \) is \( 50^\circ \). Wait, in an isosceles triangle, the base angles are equal. Wait, if \( GF = EF \), then \( \angle EGF = \angle GEF \). The sum of angles in a triangle is \( 180^\circ \). So \( \angle EGF + \angle GEF + \angle EFG = 180^\circ \). If \( \angle EFG = 50^\circ \), then \( 2\angle EGF + 50^\circ = 180^\circ \). So \( 2\angle EGF = 130^\circ \), so \( \angle EGF = 65^\circ \). Wait, but the options include 65. Wait, but the question is about \( \angle EGF \). Wait, maybe I misread. Wait, the diagram: \( \angle EFG = 50^\circ \), and \( GF = EF \) (isosceles), so base angles \( \angle EGF = \angle GEF \). So \( 180 - 50 = 130 \), divided by 2 is 65. So \( \angle EGF = 65^\circ \).

Step2: Calculate angle

Sum of angles in a triangle: \( \angle EGF + \angle GEF + \angle EFG = 180^\circ \). Since \( GF = EF \) (isosceles), \( \angle EGF = \angle GEF \). Let \( x = \angle EGF \), then \( x + x + 50^\circ = 180^\circ \). Solve: \( 2x = 130^\circ \), \( x = 65^\circ \).

Answer:

65