Question

naming arcs and finding arc measures. which statements are correct? check all that apply. arc gf is a major arc. one major arc is ghe. arc fe is a minor arc. the measure of fe is 41°. the measure of gcf is equal to the measure of ∠gde. arc ghe is a semicircle.

Explanation:

Step1: Recall arc - type definitions

A minor arc is an arc whose measure is less than 180°. A major arc is an arc whose measure is greater than 180°. A semicircle is an arc whose measure is 180°.

Step2: Analyze each statement

• Arc GF: Since the central angle ∠GDF=126° < 180°, arc GF is a minor arc, so the statement "Arc GF is a major arc" is incorrect.
• Arc GHE: The measure of arc GHE is greater than 180°, so "One major arc is $\widehat{GHE}$" is correct.
• Arc FE: Since the central angle ∠FDE = 41°<180°, "Arc FE is a minor arc" is correct.
• Measure of arc FE: Given ∠FDE = 41°, and the measure of an arc is equal to the measure of its central - angle, so "The measure of $\widehat{FE}$ is 41°" is correct.
• Measure of arc GCF: The measure of arc GCF is equal to the measure of ∠GDF + ∠FDE=126° + 41° = 167°, and ∠GDE=126°+41° = 167°, so "The measure of $\widehat{GCF}$ is equal to the measure of ∠GDE" is correct.
• Arc GHE: Since the measure of arc GHE is greater than 180°, it is not a semicircle, so "Arc GHE is a semicircle" is incorrect.

Answer:

One major arc is $\widehat{GHE}$.
Arc FE is a minor arc.
The measure of $\widehat{FE}$ is 41°.
The measure of $\widehat{GCF}$ is equal to the measure of ∠GDE.