NCERT Notes for Class 9 Maths Chapter 4 Lines and Angles
Class 9 Maths Chapter 4 Lines and Angles Notes
Chapter Name | Lines and Angles Notes |
Class | CBSE Class 9 |
Textbook Name | NCERT Mathematics Class 9 |
Related Readings |
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Lines and Angles
Point: A point is a dot made by a sharp pen or pencil. It is represented by capital letter.
Line: A straight and endless path on both the directions is called a line.
Line segment: A line segment is a straight path between two points.
Ray: A ray is a straight path which goes forever in one direction.
Angle: The space between two straight lines that diverge from a common point or between two planes that extend from a common line.
Types of Angles
1. Acute angle: An angle between 0° and 90° is called acute angle.
2. Right angle: An angle which is equal to 90° is called right angle.
3. Obtuse angle: An angle which is more than 90° but less than 180° is called obtuse angle.
4. Straight angle: An angle whose measure is 180° is called straight angle.
5. Reflex angle: An angle whose measure is between 180° and 360° is called reflex angle.
6. Complete angle: An angle which is equal to 360° is called complete angle
Pairs of Angles
1. Complementary angles: Two angles are said to be complementary if the sum of their degree measure is 90°.
For example, pair of complementary angles are 35° and 55°.
2. Supplementary angles: Two angles are said to be supplementary if the sum of their degree measure is 180°.∠AOC + ∠BOC = 180°
3. Bisector of angle: A ray which divides an angle into two equal parts is called bisector of the angle.
∠AOC = ∠BOC
4. Adjacent angles: Two angles are said to be adjacent angles if
- They have a common vertex (O)
- They have a common arm (OC)
- and their non-common arms are on either side of common arm (OA and OB).
∠AOB = ∠AOC + ∠BOC
∠AOC + ∠BOC = 180°
- Axiom 6.1: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
- Axiom 6.2: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
Theorem 6.1
To prove: If lines AB and CD mutually intersect at point O, then
(a) ∠AOC = ∠BOD (Vertically opposite angles)
(b) ∠AOD = ∠BOC
Parallel Lines
If distance between two lines is the same at each and every point on two lines, then two lines are said to be parallel.
If lines l and m do not intersect each other at any point then l || m.
Transversal line
A line is said to be transversal which intersect two or more lines at distinct points.
1. Corresponding angles
Pair of angles having different vertex but lying on same side of the transversal are called corresponding angles. Note that in each pair one is interior and other is exterior angle.
- ∠1 and ∠2
- ∠3 and ∠4
- ∠5 and ∠6
- ∠1 and ∠8
These angles are pair of corresponding angles.
2. Alternate interior angles
Pair of angles having distinct vertices and lying can either side of the transversal are called alternate interior angles.
- ∠1 and ∠2
- ∠3 and ∠4
These angles are alternate interior angles
3. Consecutive interior angles
- ∠1 and ∠2
- ∠2 and ∠4
These angles are consecutive interior angles or co-interior angles
Axiom 6.3: If two parallel lines are intersected by a transversal then each pair of corresponding angles are equal.
If AB || CD, then
- ∠PEB = ∠EFD
- ∠PEA = ∠EFC
- ∠BEF = ∠DFQ
- ∠AEF = ∠CFQ
If AB || CD, then ?
- ∠AEF = ∠EFD
- ∠BEF = ∠CFE
(i) ∠BEF + ∠DFE = 180°
(ii) ∠AEF + ∠CFE = 180°
Axiom 6.4: If two lines are intersected by a transversal and a pair of corresponding angles are equal, then two lines are parallel.
(i) If ∠PEB = ∠EFD (corresponding angles), then AB || CD
Theorem 6.4: If two lines intersected by a transversal and a pair of alternate interior angles are equal, then two lines are parallel. If ∠AEF = ∠EFD (alternate interior angles), then AB || CD.
Theorem 6.5: If two lines are intersected by a transversal and the sum of consecutive interior angles of same side of transversal is equal to 180°, the lines are parallel. If ∠AEF + ∠CFE = 180°, then AB || CD.
Theorem 6.6: Lines which are parallel to the same line are parallel to each other.
If AB || EF and CD || EF then AB || CD
Given: ΔABC
To prove: ∠A + ∠B + ∠C = 180°
Construction: Draw DE || BC
Proof: DE || BC
then ∠1 = ∠4 …(1) (alternate interior angles)
∠2 = ∠5 …(2) (alternate interior angles)
Adding equations (1) and (2),
∠1 + ∠2 = ∠4 +∠5
Adding ∠3 on both sides,
∠1 +∠2 + ∠3 = ∠3 + ∠4 + ∠5
⇒ ∠A + ∠B + ∠C = 180° (Sum of angles at a point on same side of a line is 180°)
Theorem 6.8: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Given: AABC in which, side BC is produced to D.
To Prove: ∠ACD = ∠BAC + ∠ABC
Proof: ∠ACD + ∠ACB = 180° …(1) (Linear pair)
∠ABC + ∠ACB + ∠BAC = 180° …(2)
From eqn. (1) and (2), ∠ACD + ∠ACB
= ∠ABC + ∠ACB + ∠BAC
= ∠ACD = ∠ABC + ∠BAC
