Question

which statements are correct? check all that apply. \\(\square\\) arc gf is a major arc. \\(\square\\) one major arc is \\(\overarc{ghe}\\). \\(\square\\) arc fe is a minor arc. \\(\square\\) the measure of \\(\overarc{fe}\\) is \\(41^\circ\\). \\(\square\\) the measure of \\(\overarc{gcf}\\) is equal to the measure of \\(\angle gde\\). \\(\square\\) arc ghe is a semicircle.

Explanation:

Step1: Define minor/major arcs

A minor arc spans <180°, major arc spans >180°.

Step2: Analyze Arc GF

Arc GF corresponds to central angle $\angle GDF = 126^\circ + 41^\circ = 167^\circ$, so it is a minor arc.

Step3: Analyze Arc $\overparen{GHE}$

Arc $\overparen{GHE}$ spans the rest of the circle: $360^\circ - 126^\circ = 234^\circ$, which is a major arc.

Step4: Analyze Arc FE

Arc FE corresponds to central angle $41^\circ$, so it is a minor arc.

Step5: Check $\overparen{FE}$ measure

The measure of a minor arc equals its central angle, so $m\overparen{FE}=41^\circ$.

Step6: Analyze Arc $\overparen{GCF}$

Arc $\overparen{GCF}$ is a major arc: $360^\circ - 167^\circ = 193^\circ$, while $\angle GDE = 180^\circ$, so they are not equal.

Step7: Analyze Arc GHE

Arc GHE is $234^\circ$, not $180^\circ$, so it is not a semicircle.

Answer:

• One major arc is $\overline{GHE}$.
• Arc FE is a minor arc.
• The measure of $\overline{FE}$ is $41^\circ$.