Question

which arc is congruent to \\(\widehat{eh}\\)? \\(\bigcirc\\) \\(\widehat{gh}\\) \\(\bigcirc\\) \\(\widehat{fh}\\) \\(\bigcirc\\) \\(\widehat{ge}\\) \\(\bigcirc\\) \\(\widehat{fg}\\)

Explanation:

Step1: Calculate arc $\widehat{EH}$ measure

The central angle for $\widehat{EH}$ is supplementary to the $55^\circ$ angle (since $FH$ is a straight line, $180^\circ$ total).
$\text{Measure of } \widehat{EH} = 180^\circ - 55^\circ = 125^\circ$

Step2: Calculate arc $\widehat{FG}$ measure

The central angle for $\widehat{FG}$ is supplementary to the $110^\circ$ angle (since $FH$ is a straight line, $180^\circ$ total).
$\text{Measure of } \widehat{FG} = 180^\circ - 110^\circ = 70^\circ$

Step3: Calculate arc $\widehat{GE}$ measure

Sum of all central angles in a circle is $360^\circ$.
$\text{Measure of } \widehat{GE} = 360^\circ - 55^\circ - 110^\circ - 70^\circ = 125^\circ$

Step4: Match congruent arcs

Arcs are congruent if their central angles (and thus arc measures) are equal. $\widehat{EH}$ and $\widehat{GE}$ both have a measure of $125^\circ$.

Answer:

$\boldsymbol{\overline{GE}}$