Chapter 5.4, Problem 31PPS

To determine

To Prove: The indirect proof for the theorem 5.10.

Explanation of Solution

Consider the indirect proof is that one should temporarily assume that the conclusion is false by shows the assumption as logically impossible.

Consider that $m\angle BAC>m\angle ABC$ for the diagram shown in Figure 1

  

Figure 1

The object is to prove that $BC>AC$ .

For the indirect proof consider that $BC\le AC$ . Then consider that $BC=AC$ .

Then, by the isosceles angle theorem $\angle BAC\cong \angle ABC$ . This is in contradiction to the hypothesis that $m\angle BAC>m\angle ABC$ .

Consider the case that $BC<AC$ .

For this there exists $\overline{DC}\cong \overline{BC}$ . Since, the isosceles theorem shows that $m\angle BDC\cong \angle DBC$ . Then, locate the triangle $\Delta BAD$ .

Then, Consider that the triangle corresponds to the remote interior triangle of $\angle BDC$ . Thus, the exterior angle inequality shows that $m\angle BDC>\angle BAC$

Thus,

  $\begin{array}{c}m\angle ABC=m\angle ABD+m\angle DBC>m\angle DBC\\ =m\angle BDC>\angle BAC\end{array}$

Thus, it is proved that the assumption is false and the original conclusion is true.