ANGLE RELATIONSHIPS WITH PARALLEL LINES

Find the angle x in the figure given below. Give your reason.

Example 1 :

Solution :

<ACB and <AFE are corresponding angles, so they will be equal. 

x = 125

Example 2 :

Solution :

<BCF = <GFC

Because they are alternate interior angles.

x = 57

Example 3 :

Solution :

Here x and 150 are linear pair, they add upto 180 degree.

150+ x = 180

x = 180-150

x = 30

x and y are consecutive interior angles on the same side of the transversal.

x+y = 180

30+ y = 180

y = 180 - 30

y = 150

Example 4 :

Solution :

The relationship between 99 and z :

They are corresponding angles.

So, z = 99

The relationship between z and y :

They are linear pair.

x + y = 180

99 + y = 180

y = 180 - 99

y = 81

The relationship between x and y :

They are corresponding angles.

x = y

x = 81

Example 5 :

Solution :

x + 74 = 180

x = 180 - 74

x = 106

Relationship between x and y :

They are corresponding angles, so they are equal.

y = 106

Example 6 :

Solution :

Relationship between y and 110 :

They are consecutive interior angles on the same side of the transversal.

y + 110 = 180

y = 180 - 110

y = 70

Relationship between x and 123 :

They are corresponding angles.

x = 123

Example 7 :

(a) Which angle is corresponding to angle c?

(b) Which angle is alternate to angle d?

(c) Which angle is corresponding to angle h?

(d) Which angle is vertically opposite to angle a?

(e) Which angle is alternate to angle e?

(f) Which angle is co-interior with angle c?

(g) Which angle is vertically opposite to angle h?

(h) Which angle is co-interior with angle e?

(i) Which angle is corresponding to angle a?

(j) Which angle is vertically opposite to angle g?

Solution :

(a) Corresponding angle of ∠c is ∠f.

(b) ∠d and ∠f are alternate angles.

(c) The corresponding angle of ∠h is ∠d.

(d) Vertically opposite to ∠a is ∠c.

(e) Alternate angle of ∠e is ∠c.

(f) The co-interior ∠c is ∠f.

(g) Vertically opposite to ∠h is ∠f.

(h) Co-interior angle with ∠e is ∠d.

(i) Corresponding to ∠a is ∠e.

(j) Vertically opposite to ∠g is ∠e.

Example 8 :

Find the value of the variable(s) in each figure. Explain your reasoning.

Solution :

Sum of co interior angles is 180.

15x+30 + 10x = 180 ----(1)

25x + 30 = 180

Subtract 30 on both sides.

25x = 180 - 30

25x = 150

Divide by 25 on both sides.

x = 150/25

x = 6

90 + 3y + 18 = 180 ------(2)

108 + 3y = 180

3y = 180 - 108

3y = 72

Divide by 3 on both sides.

y = 72/3

y = 24

Example 9 :

Find the value of the variable(s) in each figure. Explain your reasoning.

Solution :

Here 2x, 90 and x are linear pair. So, they add upto 180.

2x + 90 + x = 180

3x + 90 = 180

Subtract 90 on both sides.

3x = 180 - 90

3x = 90

Divide 3 on both sides.

x = 90/3

x = 30

2y and x are alternate interior angles.

2y = x

2y = 30

y = 30/2

y = 15

x and z are consecutive interior angles on the same side of the transversal.

x + z = 180

30 + z = 180

Subtract 30 on both sides.

z = 180 - 30

z = 150

Example 10 :

(Using a 3rd parallel Line – Auxilury Line)

Solution :

∠CEA = 130

Because its corresponding angle is 130 degree.

∠DEA = 100

Because alternate interior angles.

∠CEA + ∠DEA + x = 360

130 + 100 + x = 360

230 + x = 360

Subtract 230 on both sides.

x = 360 - 230

x = 130