Chapter 6.2, Problem 2PSA

To determine

To show that sides AB and CD are congruent.

Answer to Problem 2PSA

  $\overline{AB}$ and $\overline{CB}$ are congruent.

Explanation of Solution

Given information:

  $\begin{array}{l}\overleftrightarrow{PB}\perp m,\\ \angle APB\cong \angle CPB\end{array}$

Formula used:

The below properties are used:

If a line is perpendicular to a plane, it is perpendicular to every line in the plane that passes through its foot.

Perpendicular angles form right angles.

Right angles are congruent.

Two triangles are congruent by ASA congruence rule.

Corresponding parts of congruent triangles are congruent.

Proof:

It is given that,

  $\begin{array}{l}\overleftrightarrow{PB}\perp m,\\ \angle APB\cong \angle CPB\end{array}$

  

If a line is perpendicular to a plane, it is perpendicular to every line in the plane that passes through its foot.

  $\overline{BP}\perp \overline{AB}and\overline{BC}$

Perpendicular angles form right angles.

  $\angle APB and\angle CPBarerightangles.$

Right angles are congruent.

  $\angle ABP \cong \angle CBP$

By reflexive property, we get

  $\overline{BP} \cong \overline{BP}$

By ASA congruence rule, we get

  $\Delta ABP \cong \Delta CBP$

Corresponding parts of congruent triangles are congruent.

  $\overline{AB} \cong \overline{CB}$