91Ó°ÊÓ

It is given that the measure of the smallest angle in a right triangle is $45°$less than the measure of the next larger angle.

The objective is to find the measures of all three angles.

As the triangle is a right triangle so one angle measures $90°$. Now let the measure of the smallest angle be role="math" localid="1647249402193" $x°$.

It is given that the measure of the smallest angle in a right triangle is $45°$less than the measure of the next larger angle. So the measure of the next larger angle is given as$x°+45°$.

It is known that the sum of the measure of all three angles of a triangle is $180°$.

So the sum of role="math" localid="1647249507215" $x°,x°+45°,$and $90°$is equal to $80°$, and so the equation is formed as

$\begin{array}{rcl}x°+x°+45°+90°& =& 180°\\ 2x°+135°& =& 180°\\ 2x°& =& 180°-135°\\ 2x°& =& 45°\end{array}$

Divide both sides by $2$.

$\begin{array}{rcl}\frac{2x°}{2}& =& \frac{45°}{2}\\ x°& =& 22.5°\end{array}$

The smallest angle measures $x°=22.5°$.

The measure of the next larger angle is

$\begin{array}{rcl}x°+45°& =& 22.5°+45°\\ & =& 67.5°\end{array}$

And the measure of the third angle is $90°$.