You have a triangle with one angle at 100º and the other two angles have a ratio of 3:13.
You can first set variables for the 3 angles. a, b, c
a = 100 (angle you know)
b = ?
c = ?
We have 2 unknowns (b and c), so we must have at least 2 equations to solve it.
We know that the sum of all angles in any triangle is 180 degrees, so that is one equation.
Equation 1: a + b + c = 180
You also gave the ratio of 3:13 for angles b and c (the unknowns) for the other equation
Equation 2: 3/13 = b/c
Multiply both sides of the above by 'c' to get
c(3/13) = (b/c)c
The c on the right cancels and we are left with
b = 3c/13
Next, substitute b and a in the first equation
a + b + c = 180
a = 100
b = 3c/13
100 + 3c/13 + c = 180
subtract 100 from both sides
3c/13 + c = 80
multiple both sides by 13 to clear the fractions
13(3c/13) + 13c = 80(13)
3c + 13c = 1040
add left side
16c = 1040
c = 1040/16
c = 65º
You can now find b using the first or second equation. The first is easier, but best to use both to check your work.
a + b + c = 180
b = 180 - c - a
b = 180 - 65 - 100
b = 15
Second Equation Check
3/13 =b/c
let c = 65
3/13 = b/65
b = 3(65)/13 = 15 (second check)
Answer: a = 100, b = 15, c = 65
