### In a right-angled triangle show that the hypotenuse is the longest side.

1. Let $PQR$ be a right-angled triangle such that $\angle Q = 90^ \circ.$
2. As the sum of angles of a triangle is $180^ \circ$.
Now, in $\triangle PQR,$ we have \begin{aligned} & \angle P + \angle Q + \angle R = 180 ^ \circ \\ \implies & \angle P + 90 ^ \circ + \angle R = 180 ^ \circ && [As, \angle Q = 90^\circ] \\ \implies & \angle P + \angle R = 180 ^ \circ - 90 ^ \circ \\ \implies & \angle P + \angle R = 90 ^ \circ \\ \end{aligned} \\ As the measure of $\angle Q$ is equal to the sum of measures of $\angle P$ and $\angle R$, we have \begin{aligned} & \angle Q > \angle P \\ \implies & PR > QR & \ldots \text { (1) } && [\text {Side opposite to greater angle is greater.}] \\ \end{aligned} Also, \begin{aligned} & \angle Q > \angle R \\ \implies & PR > PQ & \ldots \text { (2) } && [\text{Side opposite to greater angle is greater.}] \end{aligned}
3. By equation $\text { (1) }$ and $\text { (2) }$, we have $$PR > QR \text { and } PR > PQ \\$$ As $PR$ is greater than both the sides, $\bf PR \bf$ is longest side.