Question

1. the figure shows a triangle with an angle bisector. find the measure of \\( \angle 2 \\) if \\( \angle e c d = 58 ^ { \circ } \\).

a. \\( 29 ^ { \circ } \\)
b. \\( 14.5 ^ { \circ } \\)
c. \\( 58 ^ { \circ } \\)
d. \\( 116 ^ { \circ } \\)

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal parts. So, if \(CP\) is the angle bisector of \(\angle ECD\), then \(\angle1=\angle2\) and \(\angle ECD=\angle1 + \angle2\).

Step2: Use the angle - bisector formula

Since \(\angle ECD = 58^{\circ}\) and \(\angle ECD=\angle1+\angle2\) (by angle - addition postulate) and \(\angle1 = \angle2\) (because \(CP\) is an angle bisector), we can substitute \(\angle1\) with \(\angle2\). So, \(\angle ECD=2\angle2\).

Step3: Solve for \(\angle2\)

We have the equation \(2\angle2 = 58^{\circ}\). Dividing both sides of the equation by \(2\), we get \(\angle2=\frac{58^{\circ}}{2}\).

Answer:

\(29^{\circ}\), so the answer is A.