Question

1. in figure 1, what is the sum of the measures of the angles formed by the coplanar rays with a common vertex but with no common interior points? 6. in figure 2, what is the sum of the measures of the angles formed by the radii of a circle with no common interior points? 7. in figure 2, what is the intercepted arc of ∠fab? how about ∠bac? ∠cad? ∠ead? ∠eaf? complete the table below.

central angle	measure	intercepted arc
b. ∠bac
c. ∠cad
d. ∠ead
e. ∠eaf

1. what do you think is the sum of the measures of the intercepted arcs of ∠fab, ∠bac, ∠cad, ∠ead, and ∠eaf? why? 9. what can you say about the sum of the measures of the central angles and the sum of the measures of their corresponding intercepted arcs?

Explanation:

Step1: Recall angle - sum property for coplanar rays

The sum of the measures of the angles formed by coplanar rays with a common vertex and no common interior points is \(360^{\circ}\). So, for question 5, the answer is \(360^{\circ}\).

Step2: Recall angle - sum property for circle radii

The sum of the measures of the angles formed by the radii of a circle with no common interior points is also \(360^{\circ}\) since a full - rotation around the center of a circle is \(360^{\circ}\). So, for question 6, the answer is \(360^{\circ}\).

Step3: Recall the relationship between central angles and intercepted arcs

The measure of an intercepted arc is equal to the measure of its central angle.
For \(\angle FAB\), if we assume the central angle \(\angle FAB = x_1\), then the intercepted arc \( \overset{\frown}{FB}=x_1\).
For \(\angle BAC\), if \(\angle BAC = x_2\), then the intercepted arc \( \overset{\frown}{BC}=x_2\).
For \(\angle CAD\), if \(\angle CAD = x_3\), then the intercepted arc \( \overset{\frown}{CD}=x_3\).
For \(\angle EAD\), if \(\angle EAD = x_4\), then the intercepted arc \( \overset{\frown}{ED}=x_4\).
For \(\angle EAF\), if \(\angle EAF = x_5\), then the intercepted arc \( \overset{\frown}{EF}=x_5\).

Step4: Calculate the sum of intercepted arcs

The sum of the measures of the intercepted arcs of \(\angle FAB,\angle BAC,\angle CAD,\angle EAD,\angle EAF\) is equal to the sum of the measures of these central angles. Since these central angles together form a full - rotation around the center of the circle, the sum of the measures of the intercepted arcs is \(360^{\circ}\).

Step5: State the relationship between central angles and intercepted arcs

The sum of the measures of the central angles and the sum of the measures of their corresponding intercepted arcs are equal. This is because the measure of an intercepted arc of a circle is equal to the measure of its central angle.

Answer:

1. \(360^{\circ}\)
2. \(360^{\circ}\)

7.

Central Angle	Measure	Intercepted Arc
b. \(\angle BAC\)	Measure of \(\angle BAC\)	\(\overset{\frown}{BC}\) (same measure as \(\angle BAC\))
c. \(\angle CAD\)	Measure of \(\angle CAD\)	\(\overset{\frown}{CD}\) (same measure as \(\angle CAD\))
d. \(\angle EAD\)	Measure of \(\angle EAD\)	\(\overset{\frown}{ED}\) (same measure as \(\angle EAD\))
e. \(\angle EAF\)	Measure of \(\angle EAF\)	\(\overset{\frown}{EF}\) (same measure as \(\angle EAF\))

1. \(360^{\circ}\), because the central angles corresponding to these arcs together form a full - rotation around the center of the circle.
2. They are equal, since the measure of an intercepted arc is equal to the measure of its central angle.