Question

in circle o, \\( \overline{ad} \\) and \\( \overline{be} \\) are diameters. the measure of arc ab is \\( 55^{\circ} \\) and the measure of arc cd is \\( 25^{\circ} \\). what is the measure of \\( \widehat{eac} \\)?

Explanation:

Step1: Find the measure of arc \(BC\)

Since \(AD\) and \(BE\) are diameters, \(\angle AOB=\angle DOE = 55^{\circ}\), \(\angle COD = 25^{\circ}\), and \(\angle AOB+\angle BOC+\angle COD+\angle DOE= 180^{\circ}\) (a straight - angle formed by diameter \(AD\)).
Let \(x=\angle BOC\), then \(55^{\circ}+x + 25^{\circ}+55^{\circ}=180^{\circ}\).

$$x=180^{\circ}-(55^{\circ}+25^{\circ}+55^{\circ})=45^{\circ}$$

The measure of arc \(BC\) is \(45^{\circ}\) (central - angle theorem: the measure of an arc is equal to the measure of its central angle).

Step2: Calculate the measure of arc \(EAC\)

The measure of a full - circle is \(360^{\circ}\).
The measure of arc \(EAC\) is \(m\overarc{EAC}=360^{\circ}-m\overarc{CD}-m\overarc{DE}\)
We know that \(m\overarc{CD} = 25^{\circ}\), \(m\overarc{DE}=m\overarc{AB}=55^{\circ}\) (vertical angles \(\angle AOB\) and \(\angle DOE\) are equal, so their corresponding arcs are equal)

$$m\overarc{EAC}=360^{\circ}-25^{\circ}-55^{\circ}=280^{\circ}$$

Answer:

\(280^{\circ}\)