Chapter 12.2, Problem 19E

To determine

To prove:

The third angles theorem by using the triangle sum theorem.

Explanation of Solution

Given:

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle A=\angle A_1\\ \angle B=\angle B_1\end{array}$

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.

The pairs of congruent corresponding angles

  $\begin{array}{l}\angle A=\angle A_1\\ \angle B=\angle B_1\end{array}$

By the triangle sum theorem

  $\begin{array}{l}m\angle A+m\angle B+m\angle C={180}^0\\ m\angle A_1+m\angle B_1+m\angle C_1={180}^0\end{array}$

From definition of congruent angles

  $\begin{array}{l}m\angle A=m\angle A_1\\ m\angle B=m\angle B_1\end{array}$

Substituting

  ${180}^0={180}^0$

  $m\angle A+m\angle B+m\angle C=m\angle A_1+m\angle B_1+m\angle C_1$

Substracting $m\angle A$ and $m\angle B$ from both sides

  $\begin{array}{l}m\angle C=m\angle C_1\\ \angle C\cong \angle C_1\end{array}$

Therefore,

  $\begin{array}{l}m\angle C=m\angle C_1\\ \angle C\cong \angle C_1\end{array}$