To review the basics of triangle angles, see here.
To review the definitions of complementary and supplementary angles, see here
To review proportions, see here.

Here is an example of how to convert a proportion to a math equation:

ratio of x to y is 4:7   →   x : y = 4 : 7   →   =   →   7x = 4y

Example 1: Proportions

Find the values of the following angles to the nearest degree.

A triangle has three angles labeled x, y, and z. The ratio of x to y is 3:1. The ratio of y to z is 2:1. What is the value of each of the angles, x, y, and z?

x = y = z =

Converting the proportions to math equations, we get x = 3y and y = 2z.

x + y + z	=	180	Fact: The angles of a triangle always add up to 180 degrees.
3y + y + z	=	180	substitute x = 3y
3(2z) + 2z + z	=	180	substitute y = 2z
6z + 2z + z	=	180
9z	=	180
9z ÷ 9	=	180 ÷ 9
z	=	20
y	=	2z = 2(20)	= 40
x	=	3y = 3(40)	= 120

Double check:   Does x + y + z = 180?   Yes.
x = y = z =

Example 2: Complementary angles

Find the values of the following angles to the nearest degree.

A triangle has three angles labeled x, y, and z. y and z are complementary angles and the ratio of y to z is 8:7. What is the value of each of the angles, x, y, and z?

x = y = z =

Converting the proportion to a math equation, we get 7y = 8z.

y + z	=	90	Fact: Complementary angles always add up to 90 degrees.
z + z	=	90	substitute y = z
z	=	90
z   ×	=	90   ×
z	=	42
y	=	z = (42) = 48
x	=	180 - (y + z) = 180 - (48 + 42) = 90

x = y = z =

Example 3: Two triangles and supplementary angles

Find the values of the following angles to the nearest degree.

Angles a, b, and c make up one triangle. Angles x, y, and z make up another triangle. Angles c and x are supplementary. Furthermore, a = 1 degrees, b = 7 degrees, and the ratio of x to y is 2:25. What is the value of c, x, y, and z?

c = x = y = z =

a + b + c	=	180	Fact: The angles of a triangle always add up to 180 degrees.
1 + 7 + c	=	180
8 + c - 8	=	180 - 8
c	=	172

Since angles c and x are supplementary, c + x = 180.

c + x	=	180	Fact: Supplementary angles always add up to 180 degrees.
172 + x	=	180
172 + x - 172	=	180 - 172
x	=	8

Since the ratio of x to y is 2:25, 25x = 2y. Use the value of x to find y.

25x	=	2y
25(8)	=	2y
25(8) ÷ 2	=	2y ÷ 2
100	=	y

Since we know the values of x and y, we can solve for z.

x + y + z	=	180	Fact: The angles of a triangle always add up to 180 degrees.
8 + 100 + z	=	180
108 + z - 108	=	180 - 108
z	=	72

c = x = y = z =