Question

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

Answer 1

If the given ratio is to be the ratio of the sides:
`color(white)("XXX")`No such triangle is possible.
If the given ratio is to be the ratio of the angles:
`color(white)("XXX")`The angles are `20^@, 60^@, and 100^@`

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^@`
then the ratio `1:3:5` corresponds to angles `x^@, 3x^@, and 5x^@`

Since the sum of interior angles of a triangle is `180^@`
`color(white)("XXX")x+3x+5x=180`

`color(white)("XXX")rarr 9x=180`

`color(white)("XXX")rarr x=20`

and the angles are `20^@, 60^@, and 100^@`