# Use the fact that the sum of the 3 angles in a triangle is 180^circ to answer this question. One angle in a triangle has a measure that is three times as large as the smallest angle. The measure of the third angle is 15^circ more than that of the smallest angle. Find the measure of the each of the angles.

Use the fact that the sum of the 3 angles in a triangle is ${180}^{\circ }$ to answer this question.
One angle in a triangle has a measure that is three times as large as the smallest angle. The measure of the third angle is ${15}^{\circ }$ more than that of the smallest angle. Find the measure of the each of the angles.
The SMALLEST angle has a measure of ?
The MIDDLE angle has a measure of ?
The LARGEST angle has a measure of ?
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Step 1
Let the three angles of triangle are
${x}^{\circ }=small\phantom{\rule{0ex}{0ex}}{y}^{\circ }=middle\phantom{\rule{0ex}{0ex}}{z}^{\circ }=largest$
By the given information, it is implied that
${z}^{\circ }=3{x}^{\circ }$
and ${y}^{\circ }={15}^{\circ }+{x}^{\circ }$
The sum of all angles of triangle is ${180}^{\circ }$
i.e., ${x}^{\circ }+{y}^{\circ }+{z}^{\circ }={180}^{\circ }$
$⇒{x}^{\circ }+{15}^{\circ }+{x}^{\circ }+3{x}^{\circ }={180}^{\circ }\phantom{\rule{0ex}{0ex}}⇒5{x}^{\circ }+{15}^{\circ }={180}^{\circ }\phantom{\rule{0ex}{0ex}}⇒5{x}^{\circ }={165}^{\circ }\phantom{\rule{0ex}{0ex}}{x}^{\circ }={33}^{\circ }$
Smallest angle $={33}^{\circ }$
Step 2
Middle angle $⇒{y}^{\circ }={15}^{\circ }+{x}^{\circ }$
$={15}^{\circ }+{33}^{\circ }$
$={48}^{\circ }$
Largest angle $⇒{z}^{\circ }=3{x}^{\circ }$
$=3\left({33}^{\circ }\right)$
$={99}^{\circ }$