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# Pairs of Angles – Lines & Angles

• Difficulty Level : Easy
• Last Updated : 15 Dec, 2020

When two lines share a common endpoint, called Vertex then an angle is formed between these two lines is known as the pair of angles. Below is the pictorial representation of the pair of angles.

Some of the pair of angles we saw is below:

• Complementary Angles
• Supplementary Angles
• Linear Pair of Angles
• Vertical Angles

### Complementary Angles

When we have two angles whose addition equals 90Â° then the angles are called Complementary Angles.

Example:

50Â° and  40Â° (50Â° + 40Â° = 90Â°)

70Â° and  20Â° (70Â° + 20Â° = 90Â°)

Below is the pictorial representation of the Complementary Angles.

• If we have two angles as xÂ° and  yÂ° and  xÂ° +  yÂ° =  90Â° then x is called the complementary angle of y and y is called the complementary angle of x.

Example: We have  20Â° and  70Â° then, 20Â° is a complementary angle of 70Â° and  70Â° is a complementary angle of  20Â°.

• If we have one angle as  xÂ° then to find a complementary angle we need to subtract it from  90Â°.

Example: We have  30Â° then the complementary angle of it is  90Â° –  30Â° which is  60Â°

### Supplementary Angles

When we have two angles whose addition equals to  180Â° then the angles are called Supplementary Angles.

Example:

150Â° and  30Â° (150Â° + 30Â° =  180Â°)

70Â° and  110Â° (70Â° + 110Â° =  180Â°)

Below is the pictorial representation of the Supplementary Angle.

• If we have two angles as xÂ° and yÂ° and  xÂ° +  yÂ° = 180Â° then x is called the supplementary angle of y and y is called the supplementary angle of x.

Example: We have  100Â° and  80Â° then, 100Â° is the supplementary angle of  80Â° and  80Â° supplementary angle of  100Â°.

• If we have one angle as  xÂ° then to find a supplementary angle we need to subtract it from 180Â°.

Example: We have  60Â° then the supplementary angle of it is  180Â° –  60Â° which is 120Â°

### Difference Between Complementary Angle and Supplementary Angle

When we have two angles with a common side, a common vertex without any overlap we call them Adjacent Angles.

We know what conditions two angles need to fulfill to be Adjacent angles. Let’s see some of the examples where we might get confused that whether they are adjacent angles or not.

Here  Î¸1 and  Î¸2 are having a common vertex, they don’t overlap but because they don’t share any common side they aren’t Adjacent Angles.

Here  Î¸1 and  Î¸2 are having a common vertex, they share a common side but they overlap so they aren’t Adjacent Angles.

### Linear Pair of Angles

We say two angles as linear pairs of angles if both the angles are adjacent angles with an additional condition that their non-common side makes a Straight Line.

Let’s see some examples for a better understanding of Pair of Angles.

Example 1:

Let’s call the intersection of line AC and BD to be O. Now we see four angles are there let’s try to observe them one by one.

• Î¸1 and Î¸2 are adjacent angles and their non-common sides are AO and OC, AO + OC = AC is a Straight Line so both are linear pairs of angles.
• Î¸2 and Î¸3 are adjacent angles and their non-common sides are BO and OD, BO + OD = BD is a Straight Line so both are linear pairs of angles.
• Î¸3  and Î¸ are adjacent angles and their non-common sides are CO and OA, CO + OA = CA is a Straight Line so both are linear pairs of angles.
• Î¸4  and Î¸1  are adjacent angles and their non-common sides are D0 and OB, DO + OB = DB is a Straight Line so both are linear pair of angles.

### Vertical Angles

A vertical angle is a pair of non-adjacent angles that are formed by the intersection of two Straight Lines.

Here we see line  AD and line BC intersect at one point  let’s call it X and thus four angles are formed

âˆ AXB =  Î¸1

âˆ BXD =  Î¸2

âˆ DXC =  Î¸3

âˆ CXA =  Î¸4

Î¸ and Î¸2 are non-adjacent angles and formed by the intersection of line  AD and BC therefore they are Vertical Angles are always Equal so Î¸1 = Î¸2. Similarly, Î¸3 and Î¸4 are also vertical angles therefore Î¸3 = Î¸4. Let’s try to understand with a question:

Here we see âˆ BXD  and b are vertically opposite angles therefore

b = âˆ BXD

b = 60Â°

and we also see that âˆ DXC and a are vertically opposite angles therefore

a = âˆ DXC

a = 120Â°

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