Chapter 12.3, Problem 15E

To determine

To prove:

The triangles $\Delta SPQ\cong \Delta TPQ$ are congruent using the congruence theoem.

Explanation of Solution

Given:

  $\begin{array}{l}\overline{PQ}\text{bisects}\angle SPT\\ \overline{SP}\cong \overline{TP}\end{array}$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

If two triangles are congruent.

So all the corresponding parts are congruent.

If all corresponding parts of two triangles are congruent.

So the triangle are congruent.

Two pairs of corresponding sides of the included angles of those sides are marked congruent.

By the reflexive property of congruence

  $\overline{PQ}\cong \overline{PQ}$

An angle bisect cuts an angle into two congruent

  $\begin{array}{l}\angle SPQ\cong \angle TPQ\\ \Delta SPQ\cong \Delta TPQ\end{array}$

So, the congruence theorem can be used.