Question

the circle below has center h. suppose that m∠feg = 38°. find the following. (a) m\\(\overarc{fg}\\) = \\(\square\\)° (b) m∠fhg = \\(\square\\)°

Explanation:

Step1: Recall Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Here, $\angle FEG$ is an inscribed angle intercepting arc $FG$. So, $m\angle FEG = \frac{1}{2}m\overarc{FG}$.
Given $m\angle FEG = 38^\circ$, we can solve for $m\overarc{FG}$:
$38^\circ=\frac{1}{2}m\overarc{FG}$
Multiply both sides by 2: $m\overarc{FG}=2\times38^\circ = 76^\circ$.

Step2: Recall Central Angle Theorem

The measure of a central angle is equal to the measure of its intercepted arc. $\angle FHG$ is a central angle intercepting arc $FG$, so $m\angle FHG = m\overarc{FG}$.
From Step1, $m\overarc{FG}=76^\circ$, so $m\angle FHG = 76^\circ$.

Answer:

(a) $76$
(b) $76$