Question

match the arc or central angle to the correct measure based on the figure below.

a. m\\(widehat{ifg}\\)
b. m\\(widehat{hi}\\)
c. m\\(\angle gjh\\)
d. m\\(widehat{gih}\\)

1. 50°
2. 130°
3. 230°
4. 180°

Explanation:

Step1: Recall circle - angle and arc relationships

The measure of a central angle is equal to the measure of its intercepted arc. The sum of the measures of arcs in a circle is 360°.

Step2: Analyze \(m\angle FJG\)

Since \(m\overset{\frown}{HG}=130^{\circ}\), and \(IJ\) is a diameter, \(\angle FJG\) and \(\angle HJG\) are supplementary. So \(m\angle FJG = 180 - 130=50^{\circ}\), and \(m\overset{\frown}{FG}=50^{\circ}\).

Step3: Find \(m\overset{\frown}{IF}\)

Since \(IJ\) is a diameter, \(m\overset{\frown}{IF}=180^{\circ}\)

Step4: Find \(m\overset{\frown}{HI}\)

\(m\overset{\frown}{HI}=130^{\circ}\) (because the central - angle \(\angle HJI\) and the arc \(\overset{\frown}{HI}\) have the same measure)

Step5: Find \(m\angle GJH\)

\(m\angle GJH = 130^{\circ}\) (central angle)

Step6: Find \(m\overset{\frown}{GIH}\)

\(m\overset{\frown}{GIH}=360 - 130=230^{\circ}\)

Answer:

a. \(m\overset{\frown}{IF}=180^{\circ}\) (4)
b. \(m\overset{\frown}{HI}=130^{\circ}\) (2)
c. \(m\angle GJH = 130^{\circ}\) (2)
d. \(m\overset{\frown}{GIH}=230^{\circ}\) (3)