Question

in the diagram below, $overline{ad}paralleloverline{eg}$, $angle bfc = 75^{circ}$, $angle gfc = 38^{circ}$. find $mangle fbc$. select a reason: vertical angles, linear pair (or triple), congruent angles, sum of angles in a triangle, sum of angles in a quadrilateral, base angle of an isosceles triangle, angle bisector definition, corresponding angles, alternate interior angles, alternate exterior angles, same - side interior angles, angle addition postulate, angle subtraction postulate

Explanation:

Step1: Identify the relevant angles

We know that \(\angle BFC = 75^{\circ}\) and \(\angle GFC=38^{\circ}\), and we want to find \(\angle FBC\).

Step2: Use angle - subtraction postulate

\(\angle BFC=\angle BFG+\angle GFC\). So, \(\angle BFG=\angle BFC - \angle GFC\).
Substitute the given values: \(\angle BFG = 75^{\circ}-38^{\circ}\).

Step3: Calculate the result

\(75 - 38=37^{\circ}\). But it seems there is an error in the provided answer of \(67^{\circ}\) in the image. The correct calculation using the angle - subtraction postulate gives \(\angle BFG = 37^{\circ}\).

Answer:

\(37^{\circ}\)