Question

One angle of a triangle is$30°$ more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

Short answer

The measures of all three angles are$30°,60°,and90°$.

Step-by-step solution

Step 1. Given Information

It is given that one angle of a triangle is $30°$more than the smallest angle and the largest angle is the sum of the other angles.

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

Step 2. Find and name what we are looking for

Let the measure of the smallest angle be $x°$.

Now the second angle is $30°$more than the smallest angle so it is given as $x°+30°$.

Also the largest angles is sum of the other two angles so it is given as$x°+x°+30°=2x°+30°$.

Step 3. Form and solve an equation

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

So the sum of $x°$, $x°+30°$, and $2x°+30°$is equal to $180°$

$\begin{array}{rcl}x°+x°+30°+2x°+30°& =& 180°\\ 4x°+60°& =& 180°\\ 4x°& =& 180°-60°\\ 4x°& =& 120°\end{array}$

Divide both sides by $4$.

$\begin{array}{rcl}\frac{4x°}{4}& =& \frac{120°}{4}\\ x°& =& 30°\end{array}$

Step 4. Find the three angles

The measure of the smallest angle is $30°$.

The measure of the second angle is

$\begin{array}{rcl}x°+30°& =& 30°+30°\\ & =& 60°\end{array}$

And the measure of the largest angle is

$\begin{array}{rcl}2x°+30°& =& 2\times 30°+30°\\ & =& 60°+30°\\ & =& 90°\end{array}$