Question

chords de, ef, and fg are congruent. determine the following measures. m\\(widehat{fg}=\\) m\\(\angle ecf =\\) 56.5° 67° 75°

Explanation:

Step1: Recall central - angle and arc - measure relationship

The measure of an arc is equal to the measure of its central angle.

Step2: Find \(m\widehat{FG}\)

Since the measure of the central angle corresponding to arc \(\widehat{FG}\) is \(56.5^{\circ}\), then \(m\widehat{FG}=56.5^{\circ}\).

Step3: Find \(m\angle ECF\)

The sum of the central angles in a circle is \(360^{\circ}\). Let \(m\angle ECF = x\). We know that the other central angles are \(67^{\circ}\), \(75^{\circ}\), and \(56.5^{\circ}\), and since chords \(DE\), \(EF\), and \(FG\) are congruent, the central angles corresponding to the arcs intercepted by these chords are equal.
We have \(x + 67^{\circ}+75^{\circ}+56.5^{\circ}+56.5^{\circ}=360^{\circ}\).
First, add the known angles: \(67^{\circ}+75^{\circ}+56.5^{\circ}+56.5^{\circ}=255^{\circ}\).
Then, solve for \(x\): \(x=360^{\circ}- 255^{\circ}=105^{\circ}\).

Answer:

\(m\widehat{FG}=56.5^{\circ}\)
\(m\angle ECF = 105^{\circ}\)