Question

in the figure below, \\( \angle b a d = 80 ^ { \circ } \\) and ray \\( \overrightarrow { a c } \\) bisects \\( \angle b a d \\). what is the measure of \\( \angle b a c \\)?

Explanation:

Step1: Recall the definition of an angle - bisector

An angle - bisector divides an angle into two equal parts. If ray \(\overrightarrow{AC}\) bisects \(\angle BAD\), then \(\angle BAC=\angle CAD\) and \(\angle BAD = \angle BAC+\angle CAD\).

Step2: Use the angle - bisector formula

Since \(\angle BAD = 80^{\circ}\) and \(\angle BAD=\angle BAC+\angle CAD\), and \(\angle BAC = \angle CAD\) (because \(\overrightarrow{AC}\) is the bisector of \(\angle BAD\)), we can substitute \(\angle CAD\) with \(\angle BAC\). So, \(\angle BAD=2\angle BAC\).

Step3: Solve for \(\angle BAC\)

We know that \(\angle BAD = 80^{\circ}\) and \(\angle BAD = 2\angle BAC\). Then \(\angle BAC=\frac{\angle BAD}{2}\). Substitute \(\angle BAD = 80^{\circ}\) into the formula: \(\angle BAC=\frac{80^{\circ}}{2}=40^{\circ}\).

Answer:

C. \(40^{\circ}\)