Chapter 4.5, Problem 24HP

To determine

To explain:

SSA (Side-Side-Angle) cannot be used to prove the congruenceof two triangles.

Answer to Problem 24HP

SSA (Side-Side-Angle) cannot be used to prove the congruence of two triangles.

Explanation of Solution

To find a counterexample, show a set of triangles with corresponding SSA congruence and thenshow that at least one pair of the other $2$ corresponding angles are not congruent.

If SSA was a valid congruencetheorem, then each pair of corresponding angles would be congruent.

Consider $\Delta ABC$ and $\Delta XYZ$ .

  

From the above figure,

  $\begin{array}{l}\overline{AB}\cong \overline{XY}\\ \overline{BC}\cong \overline{YZ}\\ \angle C\cong \angle Z\end{array}$

However, $\angle X\cancel{\cong }\angle A$ .

  $\angle X$ is obtuse while $\angle A$ is acute so $\Delta ABC\cancel{\cong }\Delta XYZ$ .

Thus, SSA (Side-Side-Angle) cannot be used to prove the congruence of two triangles.