Question

In triangle PQR, the measure of angle P is 36 degrees. The measure of angle Q is five times the measure of angle R. How do you find the measurement of angle Q and the measurement of angle R?

Answer 1

From https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle

Explanation:

"the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180°"

From the problem we can write:

`Q = 5R`

We can also write from Geometry, the sum of the three angles is 180 degrees, or:

`P + Q + R = 180`

We can substitute `36` for `P` and we can substitute `5R` for `Q` and solve for `R`:

`36 + 5R + R = 180`

`36 + 5R + 1R = 180`

`36 + (5 + 1)R = 180`

`36 + 6R = 180`

`-color(red)(36) + 36 + 6R = -color(red)(36) + 180`

`0 + 6R = 144`

`6R = 144`

`(6R)/color(red)(6) = 144/color(red)(6)`

`(color(red)(cancel(color(black)(6)))R)/cancel(color(red)(6)) = 24`

`R = 24`

Substituting `24` for `R` in the the equation for the relationship and calculating `Q` gives:

`Q= 5R` becomes:

`Q = 5 * 24`

`Q = 120`

Angle `Q` is 120 degrees and angle `R` is 24 degrees.