Question

what is the measure of arc ef in circle h? 41° 173° 114° 50°

Explanation:

Step1: Recall the property of a cyclic - quadrilateral

The sum of opposite angles of a cyclic - quadrilateral is 180°. In cyclic - quadrilateral DEFG, ∠D + ∠F=180° and ∠E + ∠G = 180°.

Step2: Find the measure of arc DG

The measure of an inscribed angle is half the measure of its intercepted arc. Let's first find the measure of ∠D. We know that the sum of the measures of the arcs of a circle is 360°. Let the measure of arc DG be \(x\).
We are given arc DE = 73° and arc EF = unknown, arc FG is related to the inscribed - angle information.
Since the sum of the measures of the arcs of a circle is 360°, and using the inscribed - angle property.
The inscribed angle ∠E = 123°. The arc it intercepts is arc DFG. Let the measure of arc EF be \(y\).
The measure of arc DFG=2∠E = 2×123° = 246°.
Since arc DFG=arc DE + arc EF+ arc FG and arc DE = 73°, and arc DFG = 246°.
We know that arc DFG=arc DE+ arc EF + (360°-(arc DE + arc EF))/2 (using the relationship between inscribed angles and arcs).
Another way:
The sum of the measures of the arcs of a circle is 360°. Let the measure of arc EF be \(x\).
We know that the inscribed - angle ∠E intercepts arc DFG. So arc DFG = 2∠E=246°.
Since arc DFG=arc DE + arc EF, and arc DE = 73°.
We can find arc EF as follows:
arc EF=arc DFG - arc DE.

Step3: Calculate the measure of arc EF

arc EF=246° - 73°=173°.

Answer:

173°