SOLVING PROBLEMS ON ANGLE RELATIONSHIPS

Problem 1 :

m∠EPG = 80°; m∠GPH = 30°

If ∠FPG and ∠GPH are complementary angles, then what is the measure of ∠FPG?

Solution :

m∠FPH = 90°

m∠FPG + m∠GPH = 90

m∠FPG + 30 = 90

m∠FPG = 90 - 30

m∠FPG = 60°

So, the required angle measure is 60°.

Problem 2 :

Lines AB and CD are parallel. If ∠1 measures (3x + 13)°, and ∠4 measures 151°, what is the value of x?

A. x = -4     B. x = 74     C. x = 46      D. x = 5

Solution :

∠1 = ∠4 (vertically opposite angles)

3x + 13 = 151

3x = 151 - 13

3x = 138

x = 138/3

x = 46°

So, the value of x is 46°.

Problem 3 :

Angle DOE and angle EOF are adjacent angles. If m∠DOF = 66°, m∠DOE = x, and m∠EOF = 27°, which equation could be used to solve for x?

A. x - 66° + 27° = 90°      B. x - 27° = 66°

C. x + 27° = 66°          D. x + 66° + 27° = 180°

Solution :

m∠DOF = 66°, m∠DOE = x, and m∠EOF = 27°

DOE and angle EOF are adjacent angles

m∠DOE + m∠EOF = m∠DOF

x + 27 = 66

So, option C is correct.

Problem 4 :

If ∠ROQ and ∠ROS are complementary angles, then what is the value of x and m∠ROS?

Solution :

∠QOS = ∠QOR + ∠ROS

90 = 28 + 2x

28 + 2x = 90

2x = 90 - 28

2x = 62

x = 62/2

x = 31

∠ROS = 2x

= 2(31)

 ∠ROS = 62

Problem 5 :

What is the measure of ∠GEF?

Solution :

∠DEF = 110

∠DEG + ∠GEF = 110

50 + ∠GEF = 110

∠GEF = 110 - 50

∠GEF = 60

Problem 6 :

Lines AB and CD are parallel. If ∠1 measures (2x + 15)°, and ∠4 measures 111°, what is the value of x?

Solution :

∠1 = ∠4 (Vertically opposite angles)

2x + 15 = 111

2x = 111 - 15

2x = 96

x = 96/2

x = 48

Problem 7 :

Lines AB and CD are perpendicular to each other. If ∠1 measures (4x + 3)°, and ∠2 measures 19°, what is the value of x?

Solution :

Here ∠1 and ∠2 add upto 90 degree

∠1 + ∠2 = 90

4x + 3 + 19 = 90

4x + 22 = 90

4x = 90 - 22

4x = 68

x = 68/4

x = 17

Problem 8 :

In this picture, m∠TPU = 36° and m∠UPS = 5x + 10. If ∠TPU and ∠UPS are supplementary angles, then what is the value of x?

A. 8.8      B. 134      C. 26.8      D. 30.8

Solution :

m∠TPU + m∠TPU = 180

36° + 5x + 4 = 180

5x + 40 = 180

5x = 180 - 40

5x = 140

x = 140/5

x = 28

Problem 9 :

Find the value of each variable, providing reasons for your statements:

Solution :

Reason :

Vertically opposite angles are equal.

x = 95°

Problem 10 :

Solution :

Reason :

x and 145 are linear pairs.

x + 145 = 180

x = 180 - 145

x = 35°

Problem 11 :

Solution :

Reason :

40 and x add upto 90 degree.

40 + x = 90

x = 90 - 40

x = 50°

Problem 12 :

Solution :

Reason :

50°, 60° and x are supplementary angles.

50° + 60° + x = 180°

110° + x = 180°

x = 180° - 110°

x = 70°

x, y and z are supplementary angles.

x + y + z = 180

Applying the value of x, we get

70 + y + z = 180

y + z = 110

• 50 and y are vertically opposite angles.
• 60 and z are vertically opposite angles.

So, x = 70°, y = 50° and z = 60°

Problem 13 :

Solution :

Reason :

x and y are right angles.

x = 90° and y = 90°

Problem 14 :

Solution :

Reason :

70° and x + 20° are vertically opposite angles.

70° = x + 20°

70° -  20° = x

x = 50°

Problem 15 :

Solution :

Reason :

140° and 3x - 10 are supplementary angles.

140 + 3x - 10 = 180

130 + 3x = 180

3x = 180 - 130

3x = 50

x = 50/3°

Problem 16 :

Solution :

Since the lines AB and CD are parallel, 108 and x are corresponding angles.

Reason :

x and y are supplementary.

x = 108

x + y = 180

108 + y = 180

y = 180 - 108

y = 72

Problem 17 :

Solution :

Reason :

x and 88° are vertically opposite angles.

Since the lines EF and HG are parallel, x and y are corresponding angles.

x = 88 and y = 88.