Lines and Angles Class 9

In Lines and Angles Class 9 Chapter 6, we will study the properties of the angles formed when two lines intersect each other in detail. Also, in chapter 6, we are going to discuss the properties of the angles formed when a line intersects two or more parallel lines at distinct points. In our daily life activities, we will see the different types of angles formed between the edges of plane surfaces. For obtaining a similar kind of model using the plane surfaces, one should have a thorough knowledge of angles. Lines and Angles Class 9 provides basic knowledge on different terms related to lines and angles with various theorems and axioms along with the example problems.

Lines and Angles Class 9 Topics

The topics and subtopics covered in class 9 lines and angles are given below:

  • Introduction
  • Basic Terms and Definitions
  • Intersecting Lines and Non-intersecting Lines
  • Pairs of Angles
  • Parallel Lines and a Transversal
  • Lines Parallel to the Same Line

Lines and Angles Class 9 Notes

What are the line segment and a ray?

A line segment is a portion of a line with two endpoints whereas a ray is a line with just one endpoint. Three or more points are said to be collinear points if they lie on the same straight line. In all other conditions, the points will be termed as non-collinear points.

What is an Angle?

An angle is formed when 2 rays originate from the same point. There are basically 5 types of angles:

  1. Acute Angle
  2. Obtuse Angle
  3. Right Angle
  4. Reflex Angle
  5. Straight Angle

Acute Angle: The angles measuring between 0 degrees to 90 degrees are called acute angles.

Right Angle: If an angle is exactly equal to 90 degrees then it is termed as a right angle.

Obtuse Angle: An obtuse angle is always greater than 90 degree and less than 180 degrees.

Straight Angle: The angles measuring exactly equal to 180 degrees are termed as straight angles.

Reflex Angle: A reflex angle is always greater than 180 degree but less than 360 degree.

Important Points Covered in Lines and Angles Class 9

  • A property called “Linear pair axiom” states that if a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice versa.
  • The vertically opposite angles are said to be equal if two lines intersect each other.
  • If a transversal line intersects two parallel lines, then we can say
    • each pair of corresponding angles is congruent
    • each pair of alternate interior angles is congruent,
    • each pair of interior angles on the same side of the transversal line is supplementary
  • If a transversal line intersects two lines such that either
    • any one pair of corresponding angles are similar, or
    • any one pair of alternate interior angles are similar, or
    • any one pair of interior angles on the same side of the transversal line is supplementary, then the lines are parallel.
  • Lines which are parallel to a given line are parallel to each other
  • The sum of the three angles of a triangle is equal to 180°
  • If a side of a triangle is produced, the exterior angle so formed is equivalent to the sum of the two
    interior opposite angles

Properties of Angles

  • If the sum of two angles is 90 degree then those angles are called complementary angles, whereas in the case of supplementary angles the sum of two angles is 180 degree.
  • Two angles are adjacent if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm.
  • According to Linear pair axiom, the sum of adjacent angles of a ray standing on a line is 180° and vice-versa.
  • Vertically opposite angles of lines intersecting each other are equal.
  • Lines parallel to a given line are also parallel to each other.
  • The sum of all three angles of a triangle is always 180 degree.
  • In a triangle, the exterior angle is always equal to the sum of the 2 interior opposite angles.

Lines and Angles Class 9 Examples

Example 1: 

In the given below figure, the lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Lines and Angles Class 9 Example-1

Solution:

From the diagram,

∠AOC + ∠BOE +∠COE and ∠COE +∠BOD + ∠BOE form a straight line.

we know that a straight line is equal to 180°

So, the above expression is written as:

∠AOC + ∠BOE +∠COE = ∠COE +∠BOD + ∠BOE = 180°

Now, by substituting the given values of ∠AOC + ∠BOE = 70° and ∠BOD = 40° we will get the solution,

∠COE = 110° and

∠BOE = 30°

Example 2:

In the given figure, find the values of x and y and then show that AB || CD.

Lines and Angles Class 9 Example -2

Solution:

To prove: AB || CD.

We know that a linear pair is equal to 180°.

So, x + 50° = 180°

∴ x = 130°

We also know that vertically opposite angles are equal.

So, y = 130°

In two parallel lines, the alternate interior angles are equal. In this,

x = y = 130°

This proves that alternate interior angles are equal and so, AB || CD.

Lines and Angles Class 9 Important Questions

Q.1) The lines PQ and RS in the figure intersect each other at point O. If ∠POR : ∠ROQ = 5 : 7. Find all the angles.

Lines And Angles Class 9 Notes - Chapter 6

Q.2) The ray OS stands on a line POQ. Ray OT and ray OR are angle bisectors of ∠POS and ∠SOQ, respectively. If ∠POS = x, find ∠ROT.

Lines And Angles Class 9 Notes - Chapter 6

Q.3) The lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Lines And Angles Class 9 Notes - Chapter 6

Q.4) The lines XY and MN in the given figure intersect at O. If ∠POY = 90° and a:b = 2:3, find c.

Lines And Angles Class 9 Notes - Chapter 6

 

For more solved problems and examples, stay tuned with BYJU’S – The Learning App and get detailed notes of all concepts of Class 9 mathematics.

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