Chapter 4.7, Problem 25WE

(a)

To determine

To Prove: $\overline{AD}\cong \overline{BD}$

Explanation of Solution

Given:

  

Plane $M$ is the perpendicular bisecting plane of $\overline{AB}$ .

That is, $\overline{AB}\perp$ plane $M$ and $O$ is the midpoint of $\overline{AB}$ .

Point $D$ lies on the plane $M$ which is $\perp \overline{AB}$ , and point $O$ is the midpoint of $\overline{AB}$ .

Therefore, $\overline{OD}$ is the perpendicular bisector of $\overline{AB}$ .

Point $D$ lies on the perpendicular bisector of $\overline{AB}$ , therefore it is equidistant from the endpoints of $\overline{AB}$ , $DA=DB$ , which gives $\overline{AD}\cong \overline{BD}$ .

(b)

To determine

To prove: $\overline{AC}\cong \overline{BC}$

Explanation of Solution

Given:

  

Plane $M$ is the perpendicular bisecting plane of $\overline{AB}$ .

That is, $\overline{AB}\perp$ plane $M$ and $O$ is the midpoint of $\overline{AB}$ .

Point $C$ lies on the plane $M$ which is $\perp \overline{AB}$ , and point $O$ is the midpoint of $\overline{AB}$ ,

Therefore, $\overline{OC}$ is the perpendicular bisector of $\overline{AB}$ .

Point $C$ lies on the perpendicular bisector of $\overline{AB}$ , therefore, it is equidistant from the endpoints of $\overline{AB}$ , $CA=CB$ which gives $\overline{AC}\cong \overline{BC}$ .

(c)

To determine

To Prove: $\angle CAD\cong \angle CBD$

Explanation of Solution

Consider $\Delta CAD$ and $\Delta CBD$ ,

  $\overline{AD}\cong \overline{BD}$ by part (a) and $\overline{AC}\cong \overline{BC}$ by part (b)

Further, $\overline{CD}\cong \overline{CD}$ by reflexive property.

Therefore, $\Delta CAD\cong \Delta CBD$ by Side-Side-Side-Postulate.

This gives $\angle CAD\cong \angle CBD$ corresponding angles of congruent triangles.