Circle g is inscribed with triangle e f d. point c is on the circle between points e and f. angle e is 79 degrees. the measure of arc e d is 104 degrees. what is the measure of arc ecf in circle g? 52° 98° 158° 177°
The Correct Answer and Explanation is:
To solve for the measure of arc ( \text{ECF} ) in circle ( G ), we need to understand the relationship between the given angle and the corresponding arcs in the circle.
Step 1: Understand the relationship between angles and arcs
When an angle is inscribed in a circle, the measure of the angle is half the measure of the intercepted arc. This is known as the inscribed angle theorem. In this problem, the angle ( \angle E ) is inscribed and has a measure of 79°. This angle intercepts the arc between points ( E ) and ( F ) on the circle.
Step 2: Use the inscribed angle theorem
The measure of arc ( EF ) can be determined by the relationship:
[
\text{Arc } EF = 2 \times \angle E = 2 \times 79° = 158°
]
Thus, the measure of arc ( EF ) is 158°.
Step 3: Calculate the measure of arc ( EC )
We are told that point ( C ) is on the circle between points ( E ) and ( F ), meaning arc ( EC ) is part of arc ( EF ). Therefore, the measure of arc ( EC ) is a portion of arc ( EF ).
We are also given the measure of arc ( ED ), which is 104°. The total measure of the circle is 360°, and we can find the measure of arc ( EF ) + arc ( ED ) by subtracting from 360°:
[
\text{Arc } EF + \text{Arc } ED = 360° – \text{Arc } ECF.
]
Thus, we get the measure of arc ( ECF )