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^{circ})
2. (130^{circ})
3. (230^{circ})
4. (180^{circ})

Explanation:

Step1: Recall circle - angle and arc relationships

The sum of central angles in a circle is 360°. A straight - line central angle is 180°. The measure of an arc is equal to the measure of its central angle.

Step2: Analyze the given angle

Given that one of the central angles is 130°. Since $\angle FJG$ and the 130° angle are vertical angles, $\angle FJG = 130^{\circ}$. The arc $\overset{\frown}{FG}$ has a measure equal to the measure of its central angle $\angle FJG$, so $m\overset{\frown}{FG}=130^{\circ}$.

Step3: Find the supplementary angle

The angle supplementary to the 130° angle (e.g., $\angle GJH$) is $180 - 130=50^{\circ}$ because they form a linear pair. The arc $\overset{\frown}{HI}$ which is subtended by the vertical angle of the 50° angle also has a measure of 50°.

Step4: Calculate the major arc

The major arc $\overset{\frown}{GIH}$: The whole circle is 360°. If one part of the arc (e.g., $\overset{\frown}{FG}$) is 130°, then the major arc $\overset{\frown}{GIH}=360 - 130 = 230^{\circ}$.

Answer:

a. $m\overset{\frown}{FG}=2.130^{\circ}$
b. $m\overset{\frown}{HI}=1.50^{\circ}$
c. $m\angle GJH = 1.50^{\circ}$
d. $m\overset{\frown}{GIH}=3.230^{\circ}$