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Sample Problems For Solving For Angles

Complexity=0, Mode=proportions

Find the values of the following angles to the nearest degree.
 1 A triangle has three angles labeled x, y, and z. The ratio of x to y is 2:2. The ratio of y to z is 2:6. What is the value of each of the angles, x, y, and z? x = y = z = 2 A triangle has three angles labeled x, y, and z. The ratio of x to y is 30:4. The ratio of y to z is 4:11. What is the value of each of the angles, x, y, and z? x = y = z =

Complexity=1, Mode=complementary

Find the values of the following angles to the nearest degree.
 1 A triangle has three angles labeled x, y, and z. z and x are complementary angles and the ratio of z to x is 38:7. What is the value of each of the angles, x, y, and z? x = y = z = 2 A triangle has three angles labeled x, y, and z. y and z are complementary angles and the ratio of y to z is 43:2. What is the value of each of the angles, x, y, and z? x = y = z =

Complexity=2, Mode=supplementary

Find the values of the following angles to the nearest degree.
 1 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 = 111 degrees, b = 27 degrees, and the ratio of x to y is 23:3. What is the value of c, x, y, and z? c = x = y = z = 2 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 = 39 degrees, b = 16 degrees, and the ratio of x to y is 11:9. What is the value of c, x, y, and z? c = x = y = z =

Complexity=3

Find the values of the following angles to the nearest degree.
 1 A triangle has three angles labeled x, y, and z. The ratio of x to y is 1:13. The ratio of y to z is 13:1. What is the value of each of the angles, x, y, and z? x = y = z = 2 A triangle has three angles labeled x, y, and z. x and y are complementary angles and the ratio of x to y is 7:3. What is the value of each of the angles, x, y, and z? x = y = z =

Complexity=0, Mode=proportions

Find the values of the following angles to the nearest degree.
1A triangle has three angles labeled x, y, and z. The ratio of x to y is 2:2. The ratio of y to z is 2:6.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
Write ratios in terms of n: 2n + 2n + 6n = 180 degrees
10n = 180 degrees
n = 180 ÷ 10 = 18 degrees
x = 2 × n = 2 × 18 = 36 degrees
y = 2 × n = 2 × 18 = 36 degrees
z = 6 × n = 6 × 18 = 108 degrees
2A triangle has three angles labeled x, y, and z. The ratio of x to y is 30:4. The ratio of y to z is 4:11.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
Write ratios in terms of n: 30n + 4n + 11n = 180 degrees
45n = 180 degrees
n = 180 ÷ 45 = 4 degrees
x = 30 × n = 30 × 4 = 120 degrees
y = 4 × n = 4 × 4 = 16 degrees
z = 11 × n = 11 × 4 = 44 degrees

Complexity=1, Mode=complementary

Find the values of the following angles to the nearest degree.
1A triangle has three angles labeled x, y, and z. z and x are complementary angles and the ratio of z to x is 38:7.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
z and x complementary: z + x = 90 degrees
Write z and x in terms of n for ratio: 38n + 7n = 90 degress
45n = 90 degrees
n = 90 / 45 = 2 degrees
z = 38 × n = 38 × 2 = 76 degrees
x = 7 × n = 7 × 2 = 14 degrees
y = 180 - (z + x) = 180 - 90 = 90 degrees
2A triangle has three angles labeled x, y, and z. y and z are complementary angles and the ratio of y to z is 43:2.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
y and z complementary: y + z = 90 degrees
Write y and z in terms of n for ratio: 43n + 2n = 90 degress
45n = 90 degrees
n = 90 / 45 = 2 degrees
y = 43 × n = 43 × 2 = 86 degrees
z = 2 × n = 2 × 2 = 4 degrees
x = 180 - (y + z) = 180 - 90 = 90 degrees

Complexity=2, Mode=supplementary

Find the values of the following angles to the nearest degree.
1Angles 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 = 111 degrees, b = 27 degrees, and the ratio of x to y is 23:3.
What is the value of c, x, y, and z?
c =
x =
y =
z =
Solution
a + b + c = 180 degrees
111 + 27 + c = 180 degrees
138 + c = 180 degrees
c = 42 degrees
c and x supplementary: c + x = 180 degrees
42 + x = 180 degrees
x = 138 degrees
Ratio of x to y: x:y = 23:3
x ÷ y = 23 ÷ 3
y = x × 3 ÷ 23 = 138 × 3 ÷ 23 = 18 degrees
x + y + z = 180 degrees
138 + 18 + z = 180 degrees
156 + z = 180 degrees
z = 24 degrees
2Angles 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 = 39 degrees, b = 16 degrees, and the ratio of x to y is 11:9.
What is the value of c, x, y, and z?
c =
x =
y =
z =
Solution
a + b + c = 180 degrees
39 + 16 + c = 180 degrees
55 + c = 180 degrees
c = 125 degrees
c and x supplementary: c + x = 180 degrees
125 + x = 180 degrees
x = 55 degrees
Ratio of x to y: x:y = 11:9
x ÷ y = 11 ÷ 9
y = x × 9 ÷ 11 = 55 × 9 ÷ 11 = 45 degrees
x + y + z = 180 degrees
55 + 45 + z = 180 degrees
100 + z = 180 degrees
z = 80 degrees

Complexity=3

Find the values of the following angles to the nearest degree.
1A triangle has three angles labeled x, y, and z. The ratio of x to y is 1:13. The ratio of y to z is 13:1.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
Write ratios in terms of n: 1n + 13n + 1n = 180 degrees
15n = 180 degrees
n = 180 ÷ 15 = 12 degrees
x = 1 × n = 1 × 12 = 12 degrees
y = 13 × n = 13 × 12 = 156 degrees
z = 1 × n = 1 × 12 = 12 degrees
2A triangle has three angles labeled x, y, and z. x and y are complementary angles and the ratio of x to y is 7:3.
What is the value of each of the angles, x, y, and z?
x =
y =
z =
Solution
x and y complementary: x + y = 90 degrees
Write x and y in terms of n for ratio: 7n + 3n = 90 degress
10n = 90 degrees
n = 90 / 10 = 9 degrees
x = 7 × n = 7 × 9 = 63 degrees
y = 3 × n = 3 × 9 = 27 degrees
z = 180 - (x + y) = 180 - 90 = 90 degrees

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