Chapter 6.1, Problem 7PSB

To determine

To show that sides CA and CB are congruent.

Answer to Problem 7PSB

  $\overline{CA}and\overline{CB}arecongruent.$

Explanation of Solution

Given information:

A,P and M lie in plane m .

  $\begin{array}{l}\overleftrightarrow{CP}\perp \overleftrightarrow{AP}\\ \overleftrightarrow{CP}\perp \overleftrightarrow{PB} \\ \overline{PA} \cong \overline{PB}\end{array}$

Formula used:

The below properties are used:

Perpendicular lines form right angles.

Right angles are congruent.

Two triangles are congruent by SAS congruence rule.

Proof:

It is given that,

A, P and M lie in plane m .

  $\begin{array}{l}\overleftrightarrow{CP}\perp \overleftrightarrow{AP}\\ \overleftrightarrow{CP}\perp \overleftrightarrow{PB} \\ \overline{PA} \cong \overline{PB}\end{array}$

  

Perpendicular lines form right angles.

  $\angle CPAand \angle CPBarerightangles.$

Right angles are congruent.

  $\angle CPA\cong \angle CPB$

By reflexive property, we get

  $\overline{CP} \cong \overline{CP}$

By SAS congruence rule, we get

  $\Delta CPA \cong \Delta CPB$

As corresponding parts of congruent triangles are congruent, we get

  $\overline{CA} \cong \overline{CB}$