Lines and Angles Ex 5.1 Class 7 NCERT: Concepts and Solutions

Welcome to the world of Lines and Angles! Look around you – the corner of your book, the hands of a clock, the edges of a window – they are all made of lines and angles. This chapter is the first step in understanding geometry and how shapes relate to each other. In this lesson, we will focus specifically on the concepts from lines and angles ex 5 1 class 7 ncert. We will explore special pairs of angles that have unique relationships. You'll learn what complementary and supplementary angles are, and how to find them. We will also uncover the properties of adjacent angles, linear pairs, and vertically opposite angles. By the end of this guide, you will be able to confidently identify these angle pairs and solve problems related to them, building a strong foundation for your future studies in maths.

Understanding Key Angle Relationships

Before we solve problems, let's build a strong foundation. Exercise 5.1 is all about pairs of angles that have a special connection. Let's break them down.

1. Complementary Angles:
Two angles are complementary if their sum is exactly 90°. Think of the perfect 'L' shape of a right angle. If you draw a ray from its vertex, you split it into two smaller angles. These two angles are complements of each other. For example, a 30° angle and a 60° angle are complementary because 30° + 60° = 90°.

2. Supplementary Angles:
Two angles are supplementary if their sum is exactly 180°. Imagine a straight line. If you make a point on it and draw a ray standing on it, you create two angles. These two angles are supplementary. For instance, a 110° angle and a 70° angle are supplementary because 110° + 70° = 180°.

3. Adjacent Angles:
For two angles to be adjacent, they must satisfy three conditions:

  • They have a common vertex.
  • They have a common arm (or side).
  • Their non-common arms lie on opposite sides of the common arm.

They are 'neighbours' that share a wall!

4. Linear Pair:
A linear pair is a special type of adjacent angles whose non-common arms are opposite rays, forming a straight line. Because they form a straight line (180°), a linear pair of angles is always supplementary.

5. Vertically Opposite Angles:
When two lines intersect, they form an 'X' shape. The angles that are opposite to each other at the vertex are called vertically opposite angles. A key property to remember is that vertically opposite angles are always equal to each other.

How to Find Complementary and Supplementary Angles

  1. Step 1: Identify Your Goal — First, determine whether you need to find a complementary angle (sum = 90°) or a supplementary angle (sum = 180°). Read the question carefully.
  2. Step 2: Note the Given Angle — Write down the measure of the angle that is provided in the problem. Let's say the given angle is 'x'.
  3. Step 3: Perform the Subtraction — To find the complement, calculate: 90° - x. To find the supplement, calculate: 180° - x.
  4. Step 4: State the Final Answer — The result of your subtraction is the measure of the required angle. For example, the complement of 40° is 90° - 40° = 50°. The supplement of 130° is 180° - 130° = 50°.

Important Points & Common Mistakes

  • Confusing Complementary and Supplementary: This is a very common error. Remember: 'C' for Complementary comes before 'S' for Supplementary in the alphabet, just as 90° comes before 180°. Use this trick to keep them straight.
  • All Adjacent Angles are NOT a Linear Pair: Adjacent angles only form a linear pair if their non-common sides form a straight line. If they don't, the angles are just adjacent and their sum is not necessarily 180°.
  • Assuming any two angles that add to 180° form a linear pair: For angles to form a linear pair, they MUST be adjacent. Two separate angles, like a 100° angle in one corner of a room and an 80° angle in another, are supplementary but do not form a linear pair.
  • Vertically Opposite vs. Adjacent: When two lines intersect, an angle and the one next to it are adjacent (and form a linear pair). The angle across the vertex from it is its vertically opposite angle.

Practice Questions with Solutions

  • Q: Find the complement of the angle 63°. A: Step 1: We know that two angles are complementary if their sum is 90°. Step 2: Let the required complementary angle be 'x'. Then, x + 63° = 90°. Step 3: To find x, we subtract 63° from 90°. So, x = 90° - 63° = 27°. Final answer: The complement of 63° is 27°.
  • Q: Find the angle which is equal to its supplement. A: Step 1: We know that supplementary angles add up to 180°. Step 2: Let the angle be 'x'. According to the question, its supplement is also 'x'. Step 3: Therefore, x + x = 180°. This simplifies to 2x = 180°. Step 4: Solving for x, we get x = 180° / 2 = 90°. Final answer: The angle which is equal to its supplement is 90°.
  • Q: Two lines PQ and RS intersect at point O. If ∠POR = 50°, find the measures of ∠POS, ∠SOQ, and ∠ROQ. A: Step 1: Identify the relationships. ∠POR and ∠SOQ are vertically opposite angles. ∠POR and ∠POS form a linear pair. ∠POS and ∠ROQ are vertically opposite angles. Step 2: Find ∠SOQ. Since vertically opposite angles are equal, ∠SOQ = ∠POR = 50°. Step 3: Find ∠POS. Since ∠POR and ∠POS form a linear pair, their sum is 180°. So, ∠POS + 50° = 180°. This gives ∠POS = 180° - 50° = 130°. Step 4: Find ∠ROQ. Since ∠POS and ∠ROQ are vertically opposite angles, they are equal. Therefore, ∠ROQ = ∠POS = 130°. Final answer: ∠POS = 130°, ∠SOQ = 50°, and ∠ROQ = 130°.
  • Q: Can two obtuse angles be supplementary? A: Step 1: An obtuse angle is an angle greater than 90° but less than 180°. Step 2: Let's take the smallest possible integer obtuse angles, for example, 91° and 91°. Step 3: Their sum is 91° + 91° = 182°. This sum is greater than 180°. Step 4: Since the sum of any two obtuse angles will always be greater than 180°, they cannot be supplementary. Final answer: No, two obtuse angles cannot be supplementary.

Frequently Asked Questions

What is the main difference between supplementary angles and a linear pair?

A linear pair must be adjacent angles whose non-common sides form a straight line. All linear pairs are supplementary (add to 180°). However, two angles can be supplementary without being adjacent (e.g., a 120° angle and a 60° angle in different parts of a diagram).

How can I remember the difference between complementary and supplementary?

A simple trick is to use the alphabet. 'C' for Complementary comes before 'S' for Supplementary. Similarly, the number 90 comes before 180. So, Complementary = 90° and Supplementary = 180°.

Are vertically opposite angles always equal?

Yes, always. When any two straight lines intersect, the angles that are directly opposite each other at the point of intersection (the vertex) are always equal in measure. This is a fundamental theorem in geometry.