Question

match the arc or central angle to the correct measure based on the figure below. a. m\\(widehat{fg}\\) 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 sum of angles around a point is 360°. If one arc has a measure of 130°, and we assume the circle is centered at \(J\).

Step2: Find \(m\angle FJG\)

Since the given arc \(HG\) has a measure of 130°, and \(\angle FJG\) and \(\angle HJG\) are supplementary (a straight - line through the center of a circle forms a 180° angle at the center), if \(\angle HJG = 130^{\circ}\), then \(\angle FJG=180 - 130=50^{\circ}\).

Step3: Determine arc measures

• \(m\widehat{FG}\): The measure of an arc is equal to the measure of its central angle. So \(m\widehat{FG}=50^{\circ}\) (since the central angle \(\angle FJG = 50^{\circ}\)).
• \(m\widehat{HI}\): The arc \(\widehat{HI}\) is a semi - circle, so \(m\widehat{HI}=180^{\circ}\).
• \(m\angle GJH\): Given in the figure, \(m\angle GJH = 130^{\circ}\).
• \(m\widehat{GIH}\): The measure of the major arc \(m\widehat{GIH}=360 - 130=230^{\circ}\).

Answer:

a. 1. \(50^{\circ}\)
b. 4. \(180^{\circ}\)
c. 2. \(130^{\circ}\)
d. 3. \(230^{\circ}\)