Exercise 28 Page 378

Let's mark the angles we are asked to compare. We will use the same color for the opposite sides in the triangles.

We are asked to use an indirect argument to prove Theorem $5.14,$ the Converse of the Hinge Theorem.

Step $1$

To start an indirect proof, we assume that the statement is not true.

Step $2$

Next, we look at consequences of our assumption until we find a contradiction. Keep in mind that in triangles $△RST$ and $△UWV$ two pairs of sides are congruent. The assumption $m\angle S\le m\angle W$ is true either when $m\angle S=m\angle W$ or when $m\angle S<m\angle W.$ We should consider these two cases separately.

Case $1$: $m\angle S=m\angle W$

If $m\angle S=m\angle W,$ then in triangles $△RST$ and $△UWV$ two pair of sides and the included angles are congruent. According to the Side-Angle-Side (SAS) Congruence Theorem this means that the two triangles are congruent. Corresponding sides of congruent triangles are congruent, so this would imply $RT=UV.$ However, it is given that $RT>UV,$ so this is a contradiction.

Case $2$: $m\angle S<m\angle W$

Since in triangles $△RST$ and $△UWV$ two pair of sides are congruent, we can use the Hinge Theorem (Theorem $5.13$ of the book). It is given that $RT>UV,$ so this is again a contradiction.

Step $3$

Since in both cases we arrived at a contradiction, this means that our assumption is not true. Angle $\angle S$ must be larger than $\angle W.$ This proves the Converse of the Hinge Theorem.