Chapter 6.3, Problem 16WE

To determine

To prove: that if two angles of a triangle are not congruent, then the sides opposite those are not congruent.

Answer to Problem 16WE

Explanation of Solution

Given information:

Calculation:

Consider a $\Delta ABC$

Such that $\angle ABC\cancel{\cong }\angle ACB$ .

Use an indirect proof, to prove that $AC\cancel{\cong }AB$ as follows:

An indirect proof is initiated by assuming temporarily that whatever is need to prove is untrue and then work from there to finally conclude that the assumption is untrue.

Proof:

Assume temporarily that $AC\cong AB$ .

This would make this triangle an isosceles triangle.

From the definition of an isosceles triangle, its base angles are equal.

Hence, $\angle ABC\cong \angle ACB$ .

However, this contradicts what is given, that $\angle ABC\cancel{\cong }\angle ACB$ .

Therefore, the assumption, that $AC\cong AB$ is not true.

Therefore, $AC\cancel{\cong }AB$ .