# How do you find all the measures of the angles in a triangle that have the ratio of 1:3:5?

Mar 6, 2017

If the given ratio is to be the ratio of the sides:
$\textcolor{w h i t e}{\text{XXX}}$No such triangle is possible.
If the given ratio is to be the ratio of the angles:
$\textcolor{w h i t e}{\text{XXX}}$The angles are ${20}^{\circ} , {60}^{\circ} , \mathmr{and} {100}^{\circ}$

#### Explanation:

Case 1: ratio is ratio of lengths of sides
The length of the two shorter sides of a triangle must be greater than the length of the third side.

Case 2: ratio is ratio of angles
If the smallest angle is ${x}^{\circ}$
then the ratio $1 : 3 : 5$ corresponds to angles ${x}^{\circ} , 3 {x}^{\circ} , \mathmr{and} 5 {x}^{\circ}$

Since the sum of interior angles of a triangle is ${180}^{\circ}$
$\textcolor{w h i t e}{\text{XXX}} x + 3 x + 5 x = 180$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 9 x = 180$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow x = 20$

and the angles are ${20}^{\circ} , {60}^{\circ} , \mathmr{and} {100}^{\circ}$