Pairs of angles is an important concept of geometry. Pairs of angles are a foundation topic for many advanced topics. The real-life applications of angles are everywhere, from the corners of our homes to the layout of our streets. It is necessary to understand how angles are related to one another. In this article, we are going to learn about different pairs of angles and important formulas/concepts related to pairs of angles and will solve different types of examples based on them.

What are Pairs of Angles?

Pairs of Angles is the specific relationship between two pairs of angles. This relationship can be because of different reasons such as based on their measures, their positions relative to each other, or the lines and shapes they form or interact with. Some main types of pairs of angles are complementary angles, supplementary angles, linear pair angles, adjacent angles, vertical angles, etc.

Types of pairs of Angles

There are different types of pairs of angles. Some of the important pairs are mentioned below:

• Complementary Angles: When the sum of angles adds up to 90 degrees. These angles are known as Complementary Angles.

• Supplementary Angles: When the sum of angles adds up to 180 degrees. These angles are known as Supplementary Angles.

• Vertical Angles: If two angles are vertical angles, then they are equal: ∠A = ∠B

• Linear Pair of Angles: A pair of adjacent angles formed when two lines intersect each other at a single point and form a straight line. The sum of angles of a linear pair is always equal to 180°.

Important Related Formulas / Concepts

There are different formulas and concepts related to pairs of angles. These formulas are important to understand and learn. Some are mentioned below:

Complementary Angles

If two angles ∠A and ∠B are complementary, then their sum is 90.

∠A + ∠B = 90°

Supplementary Angles

If two angles ∠A and ∠B are supplementary, then their sum is 180 degrees.

∠A + ∠B = 180°

Linear Pair of Angles

If two angles ∠A and ∠B form a linear pair, then their sum is 180 degrees.

∠A + ∠B = 180°

Angle Sum Property of a Triangle

Angle Sum Property of a Triangle states that, the sum of all three interior angles in a triangle is always 180 degrees.

∠A + ∠B + ∠C = 180°

Angles in a Quadrilateral

The sum of the angles in a quadrilateral is always 360 degrees.

∠A + ∠B + ∠C + ∠D = 360°

Adjacent Angles in a Parallelogram

Adjacent angles in a parallelogram are supplementary.

∠A + ∠B = 180° and ∠C + ∠D = 180°

Angles in a Regular Polygon

The measure of each interior angle of a regular polygon with n sides is given by:

Interior Angle = (n -2) × 180°/n

The measure of each exterior angle of a regular polygon with n sides is given by:

Exterior Angle = 360°/n

Pairs of Angles Practice Questions

Examples 1: If two angles are complementary. If one of the angles is 40°, then find another angle.

Solution:

Condition for two angles are complementary: ∠A + ∠B = 90°

∠A + ∠B = 90°

given: ∠A = 40°, ∠B = ?

∠A + ∠B = 90°

40 + ∠B = 90°

∠B = 90° – 40°

∠B = 50°

Examples 2: If two angles are supplementary and the measure of one of the angles is 110°, then find another angle.

Solution:

Condition for two angles are complementary:

∠A + ∠B = 180°

∠A + ∠B

given: ∠A = 110°, ∠B = ?

∠A + ∠B = 180°

110° + ∠B = 180°

∠B = 180° – 110°

∠B = 70°

Examples 3: ∠E and ∠F are vertical angles. If a measure of ∠E is 75°, find measure of ∠F.

Solution:

By vertical angle property, we know that vertical angles are equal:

∠E = ∠F

Given, ∠E = 75°,

∠F = 75°

So, ∠F = 75°

Examples 4: ∠G and ∠H form a linear pair. If ∠G is 130°, find ∠H.

Solution:

We know that sum of angles forming a linear pair is 180

∠G + ∠H = 180°

Given, ∠G = 130°,

130° + ∠H = 180°

∠H = 180° − 130°

= 50°

Examples 5: Two angles are adjacent and one of them is 35°. If their sum is 90°, find the other angle.

Solution:

Let us take the measure of another angle be ∠X.

Given, ∠X + 35° = 90°,

∠X = 90° − 35° = 55°

So, the measure of other angle is 55°.

Examples 6: Find the measure of each interior angle of a regular hexagon.

Solution:

By formula we know that for a regular polygon with n sides,

Interior Angle = ( n – 2) × 180°/n

For a hexagon, number of side = 6

So, Interior Angle = (6 – 2) × 180°/6

= 4 × 180°/6

= 720°/6

= 120°

So, each interior angle of a regular hexagon is 120°.

Examples 7: In a parallelogram, one of the angles is 110°. Find the adjacent angle.

Solution:

We know that adjacent angles in a parallelogram are supplementary,

∠A + ∠B = 180°

given angle ∠A = 110°,

110° + ∠B = 180°

∠B = 180° − 110° = 70°

Examples 8: If two angles are complementary. If one of the angles is 30°, then find another angle.

Solution:

Condition for two angles are complementary:

∠A + ∠B = 90

∠A + ∠B = 90°

given: ∠A = 30° ∠B = ?

∠A + ∠B = 90°

30 + ∠B = 90°

∠B = 90° – 30°

∠B = 60°

Examples 9: If two angles are supplementary and the measure of one of the angles is 130°, then find another angle.

Solution:

Condition the for two angles are complementary:

∠A + ∠B = 180°

∠A + ∠B

given: ∠A = 130° ∠B = ?

∠A + ∠B = 180°

130° + ∠B = 180°

∠B = 180° – 130°

∠B = 50°

Worksheet: Pair of Angles

Q1. If two angles are complementary and one of them is 45°, find the other angle.

Q2. If two angles are supplementary and the the measure of one of the angles is 90°, find the other angle.

Q3. If ∠X and ∠Y are vertical angles. If the measure of ∠X is 85°, find the measure of ∠Y.

Q4. If ∠P and ∠Q form a linear pair. If ∠P is 150°, find ∠Q.

Q5. Two angles are adjacent and one of them is 25°. If their sum is 90°, find the other angle.

Q6. Find the measure of each interior angle of a regular octagon.

Q7. In a parallelogram, one of the angles is 100°. Find the adjacent angle.

Q8. If two angles are complementary and one of the angles is 20°, find the other angle.

Q9. If two angles are supplementary and the and measure of one of the angles is 140°, find the other angle.

Q10. In a figure, ∠M and ∠N are alternate interior angles. If ∠M is 60°, find ∠N.

Answer Key

1. Other Angle = 45°
2. Other Angle = 90°
3. ∠Y = 85°
4. ∠Q = 30°
5. ∠X = 65°
6. Interior Angle = 135°
7. Adjacent Angle = 80°
8. Other Angle = 70°
9. Other Angle = 40°
10. ∠N = 60°

Conclusion

It is important to understand pairs of angles and their relationships for solving many geometric problems. After Practicing questions on complementary, supplementary, vertical, and adjacent angles, and a linear angles we can enhance our problem-solving skills. The concept of pairs of angles has various real-world applications where precise angle measurements and relationships are important.

Also Read:

• Angles Definition
• Pairs of Angles
• Types of Angles

Frequently Asked Questions

What are complementary angles?

Complementary angles are two angles whose sum is equal to 90 degrees. For example, if one angle measures 30 degrees, its complementary angle measures 60 degrees.

What are supplementary angles?

Supplementary angles are two angles whose sum is equal to 180 degrees. For example, if one angle measures 110 degrees, its supplementary angle measures 70 degrees.

What are linear pairs of angles?

Linear pairs are adjacent angles formed when two lines intersect. The sum of angles of a linear pair is equal to 180 degrees.

Why is it important to practice questions on pairs of angles?

Practicing questions on pairs of angles helps to learn the concepts, improves problem-solving skills, and helps to build foundation for complex geometric problems.

How are pairs of angles used in real-life applications?

Pairs of angles have many real-life applications in different fields such as engineering, architecture, and carpentry for designing structures, ensuring accuracy in measurements, and creating specific angle requirements.