**Angle Relationships :**

Two or more angles can be related if certain conditions are met. Here, we will learn different types of angle relationships.

(i) Congruent Angles

(ii) Vertical Angles

(iii) Complementary Angles

(iv) Supplementary Angles

(v) Adjacent Angles

(vi) Adjacent Angles in Parallelogram

(vii) Linear Pair

(viii) Corresponding Angles

(ix) Alternate Interior Angles

(x) Alternate Exterior Angles

(xi) Consecutive Interior Angles

(xii) Consecutive Exterior Angles

Let us see the above different types of angle relationships in detail.

**(i) Congruent angles : **

Congruent angles are angles that have the same measure.

**(ii) Vertical angles : **

Vertical angles have a common vertex, but they are never adjacent angles. And also, vertical angles are always congruent.

**(iii) Complementary angles : **

If the sum of two angles is 90⁰, then those two angles are called as complementary angles.

**(iv) Supplementary angles : **

If the sum of two angles is 180⁰, then those two angles are called as supplementary angles.

**(v) Adjacent angles : **

Adjacent angles are two angles that have a common vertex and a common side.

**(vi) Consecutive angles in Parallelogram: **

Any two consecutive angles of a parallelogram are supplementary.

**(vii) Linear Pair : **

A linear pair is a pair of adjacent angles formed when two lines intersect.

The two angles of a linear pair are always supplementary, which means their measures add up to 180°.

To know about Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Consecutive Interior Angles and Consecutive Exterior Angles in detail,

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**Question 1 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is very clear that the angle measures (6x + 4)° and (4x + 6)° are complementary.

So, we have

(6x + 4)° + (4x + 6)° = 90°

6x + 4 + 4x + 6 = 90

10x + 10 = 90

Subtract 10 from both sides.

10x = 80

Divide both sides by 10.

x = 8

Hence, the value of "x" is 8.

**Question 2 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is clear that the angle measures (4x + 7)° and (6x + 3)° are complementary.

So, we have

(4x + 7)° + (6x + 3)° = 90°

4x + 7 + 6x + 3 = 90

10x + 10 = 90

Subtract 10 from both sides.

10x = 80

Divide both sides by 10.

x = 8

Hence, the value of "x" is 8.

**Question 3 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is clear that (2x+3)° and (x-6)° are supplementary angles.

So, we have

(2x + 3)° + (x - 6)° = 180°

2x + 3 + x - 6 = 180

3x - 3 = 180

3x = 183

x = 61

Hence the value of "x" is 61.

**Question 4 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is clear that (5x+4)°, (x-2)° and (3x+7)° are supplementary angles.

So, we have

(5x + 4)° + (x - 2)° + (3x + 7)° = 180°

5x + 4 + x -2 + 3x + 7 = 180

9x + 9 = 180

9x = 171

x = 19

Hence the value of "x" is 19.

**Question 5 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is clear that (3x+7)° and 100° are vertical angles.

Because (3x+7)° and 100° are vertical angles, they are congruent.

So, we have

(3x + 7)° = 100°

3x + 7 = 100

Subtract 7 from both sides.

3x = 93

Divide both sides by 3.

x = 31

Hence the value of "x" is 31.

**Question 6 : **

Find the value of "x" in the diagram given below.

**Solution :**

In the diagram above, it is clear that (x + 33)° and 98° form a linear pair.

Because the two angles of a linear pair are always supplementary, we have

(x + 33)° + 98° = 180°

x + 33 + 98 = 180

x + 131 = 180

Subtract 131 from both sides.

x = 49

Hence the value of "x" is 31.

**Question 7 : **

In the diagram given below, l_{1 }lines l_{2} are parallel and t is a transversal. Find the value of "x".

In the above diagram, (2x + 20)° and (3x - 10)° are corresponding angles.

When two parallel lines are cut by a transversal, corresponding angles are congruent.

So, we have

(2x + 20)° = (3x - 10)°

2x + 20 = 3x - 10

30 = x

Hence, the value of "x" is 30.

**Question 8 : **

In the diagram given below, l_{1 }lines l_{2} are parallel and t is a transversal. Find the value of "x".

In the above diagram, (2x + 10)° and (x + 5)° are consecutive interior angles.

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary.

So, we have

(2x + 10)° + (x + 5)° = 180°

2x + 10 + x + 5 = 180

3x + 15 = 180

Subtract 15 from both sides.

3x = 165

Divide both sides by 3.

x = 55

Hence, the value of "x" is 55.

**Question 9 : **

In the diagram given below, a_{ }lines b are parallel and t is a transversal. Find the value of "x".

In the diagram diagram, (2x + 26)° and (3x - 33)° are alternate interior angles.

When two parallel lines are cut by a transversal, alternate interior angles are congruent.

So, we have

(2x + 26)° = (3x - 33)°

2x + 26 = 3x - 33

59 = x

Hence, the value of "x" is 59.

**Question 10 : **

In the diagram given below, find the value of "x".

**Solution : **

In the diagram diagram, it is clear that AB||CD and AD||BC.

So ABCD is a parallelogram.

In a parallelogram, two consecutive angles are always supplementary.

Then, we have

x° + (2x)° = 180°

x + 2x = 180

3x = 180

Divide both sides by 3.

x = 60

Hence, the value of "x" is 60.

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