PROBLEMS ON ALTERNATE ANGLES

Find the value of unknown of the following :

Problem 1 :

Solution :

Considering the angles marked in the picture given above are in alternate position, they are alternate opposite angles.

x + 10 = 2x - 25

Subtracting 2x on both sides.

x - 2x + 10 = -25

Subtracting 10 on both sides.

-x = -25 - 10

-x = -35

x = 35

Problem 2 :

Solution :

x = 61° (Alternate interior angles)

Problem 3 :

Solution :

21x + 6 and 22x + 2 are alternate interior angles.

21x + 6 = 22x + 2

Subtracting 22x on both sides.

21x - 22x + 6 = 2

Subtracting 6 on both sides.

-x = 2 - 6

x = 4

Problem 4 :

Solution :

12x + 8 and 13x + 2 are alternate interior angles.

12x + 8 = 13x + 2

Subtracting 13x on both sides.

12x - 13x + 8 = 2

Subtracting 8 on both sides.

-x = 2 - 8

x = 6

Problem 5 :

Solution :

28x + 3 and 29x - 1 are alternate interior angles.

28x + 3 = 29x - 1

Subtracting 29x on both sides.

28x - 29x + 3 = -1

Subtracting 3 on both sides.

-x = -1 - 3

x = 4

Problem 6 :

Solution :

19x - 3 and 130 are alternate interior angles.

19x - 3 = 130

Add 3 on both sides.

19x = 130 + 3

19x = 133

Dividing by 19 on both sides.

x = 133/19

x = 7

Problem 7 :

Solution :

x + 146 and 135 are alternate interior angles.

x + 146 = 135

x = 135 - 146

x = -11

Problem 8 :

Solution :

14x - 2 and 110 are alternate exterior angles.

14x - 2 = 110

Adding 2 on both sides.

14x = 110 + 2

14x = 112

Dividing by 14 on both sides.

x = 112/14

x = 8

Problem 9 :

Solution :

11x - 7 and 125 are alternate exterior angles.

11x - 7 = 125

Adding 7 on both sides.

11x = 125 + 7

11x = 132

Dividing by 11 on both sides.

x = 132/11

x = 12

Problem 10 :

Solution :

14x + 4 and 13x + 13 are alternate exterior angles.

14x + 4 = 13x + 13

Subtracting 13x on both sides.

14x - 13x + 4 = 13

x + 4 = 13

Subtracting 4 on both sides.

x = 13 - 4

x = 9

Problem 11 :

Copy and complete the statement. List all possible correct answers.

a) ∠BCG and ____ are corresponding angles.

b) ∠BCG and ____ are consecutive interior angles.

c) ∠FCJ and ____ are alternate interior angles.

d) ∠FCA and ____ are alternate exterior angles.

Solution :

a) ∠BCG and ∠HJG are corresponding angles.

b) ∠BCG and ∠CJH are consecutive interior angles.

c) ∠FCJ and ∠DFC are alternate interior angles.

d) ∠FCA and ∠GJH are alternate exterior angles.

Problem 12 :

classify the angle pair as corresponding, alternate interior, alternate exterior, or consecutive interior angles.

a) ∠5 and ∠1

b) ∠11 and ∠13

c) ∠6 and ∠13

d) ∠2 and ∠11

Solution :

a) ∠5 and ∠1 are at the same position, then they corresponding angles.

b) ∠11 and ∠13 are inside on the same side of the transversal, then they are known as co-interior or consecutive interior angles.

c) ∠6 and ∠13 are inside on the same side of the transversal, then they are known as co-interior or consecutive interior angles.

d) ∠2 and ∠11 are alternate interior angles.

Problem 13 :

Identify all pairs of angles of the given type.

a. corresponding

b. alternate interior

c. alternate exterior

d. consecutive interior

Solution :

a. corresponding

• ∠1 and ∠5
• ∠2 and ∠6
• ∠3 and ∠7
• ∠4 and ∠8

These are corresponding angles.

b. alternate interior

• ∠4 and ∠5
• ∠2 and ∠7

These are alternate interior angles

c. alternate exterior

• ∠3 and ∠6
• ∠1 and ∠8

These are alternate exterior angles

d. consecutive interior

• ∠4 and ∠7
• ∠2 and ∠5

Problem 14 :

a) What do you notice about the following angle pairs?

x° and y°

y° and z°

x° and z°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

x =

y =

z =

w =

v =

Explain how you obtained each answer.

Solution :

a) 

x° and y° are linear pair of angles, then they add upto 180 degree.

y° and z° are linear pair of angles, then they add upto 180 degree.

x° and z° are vertical angles, they must be equal.

b)

x = 180 - 108 ==> 72

y = 108 (Vertical angles)

z = 180 - 108 ==> 72

w = 72(corresponding angles)

v = 180 - (z + w)

= 180 - (72 + 72)

= 180 - 144

= 36

Problem 15 :

What do you notice about the following angle pairs?

a° and b°

c° and d°

c° and e°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

c =

d =

e =

Explain how you obtained each answer.

Solution :

a) 

• a° and b° are complementary angles
• c° and d° are supplementary angles.
• c° and e° are vertical angles.

b)

• c = 90
• d = 90
• e = 90