Question

In a triangle RSM, the measure of angle SRM is twice the measure of angle RSM. A point O is selected on side RS such that SO=SM. The length of the angle bisector of angle RMS equals RO. What is the degree measure of angle RSM. The answer is a positive integer.

Answer 1

Answer:
`36^(\circ)`

Explanation:

Given

`\angle RSM =2\angle SRM`

suppose
`\angle RSM =\angle 2`

and
`\angle SRM=1`

So from figure

`\angle SOM =\angle OMS=\angle 3`

and
`\angle OMS=\angle 0MR`

In triangle
`SOM`

`\angle 2+\angle 3+\angle 3=180^(\circ)`

`\angle 2+2\angle 3=180^(\circ)\quad \ldots(i)`

In triangle
`RSM`

`\angle 1+\angle 2+\angle 3=180^(\circ)\quad \ldots(ii)`

and
`\angle 1=2\angle 2`

Using this and Substitute this value in
`(ii)`

`2\angle 2+\angle 2+\angle 3=180^(\circ)`

`3\angle 2+\angle 3=180^(\circ)\quad \ldots(iii)`

Solving (i) and (iii) we get

`2\angle 2=\angle 3`

Substitute in equation (i) we get

So
`\angle 2=(180)/(5)`

`\angle 2=36^(\circ)`

So
`\angle RSM=\angle 2=36^(\circ)`