Exercise 35 Page 48

Let's begin by reviewing the angle bisector construction. Having an angle $\angle A,$ we put the compass point on vertex $A$ and draw an arc that intersects the sides of $\angle A.$ We can label the points of intersection $B$ and $C.$

Now we put the compass point on $C$ and draw an arc. With the same compass setting, we draw an arc using point $B$ and label the point of intersection as $D.$ Finally, we can draw the ray $\overrightarrow{AD}$ which is the angle bisector of $\angle A.$

Since the arcs from the points $B$ and $C$ were drawn with the same compass setting, we can tell that the segments $\overline{BD}$ and $\overline{CD}$ are congruent. Also, the segments $\overline{AB}$ and $\overline{AC}$ were drawn with the same compass setting, so they are congruent. As we can see, the triangles $ABD$ and $ACD$ share the side $AD,$ so $\overline{BD}\cong \overline{CD}$ and $\overline{AB}\cong \overline{AC}.$ Therefore, these triangles are congruent! From the exercise we know that if each side of one triangle is congruent to a side of the other triangle, then we can conclude that the triangles are congruent without finding the angles. Thus, the angles in our triangles are congruent. In particular we can say that the angles $\angle BAD$ and $\angle CAD$ are congruent. Therefore, the ray $\overrightarrow{AD}$ is an angle bisector of $\angle BAC.$