Question

in the diagram below, $overline{ad} parallel overline{eg}$, $\text{m}angle bfc = 75^circ$ and $\text{m}angle gfc = 38^circ$. find $\text{m}angle fbc$.

angle	reason
$\text{m}angle gfc = 38^circ$	given
$angle square = square^circ$	select a reason

(diagram: e---f---g (with $angle gfc = 38^circ$, $angle bfc = 75^circ$), a---b---c---d, lines $ad parallel eg$; you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.)

Explanation:

Step1: Calculate ∠BFG

First, find the measure of ∠BFG by adding the given adjacent angles.
$\text{m}\angle BFG = \text{m}\angle BFC + \text{m}\angle GFC = 75^\circ + 38^\circ = 113^\circ$

Step2: Use parallel line property

Since $\overline{AD} \parallel \overline{EG}$, consecutive interior angles are supplementary. So $\text{m}\angle FBC + \text{m}\angle BFG = 180^\circ$.
$\text{m}\angle FBC = 180^\circ - \text{m}\angle BFG$

Step3: Substitute value and solve

Plug in the value of ∠BFG to find ∠FBC.
$\text{m}\angle FBC = 180^\circ - 113^\circ = 67^\circ$

Answer:

$\text{m}\angle FBC = 67^\circ$
(Reason: Consecutive interior angles of parallel lines are supplementary)