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NCERT Class 9 Mathematics Revision Notes Chapter 6 Lines and Angles
Extramarks has provided Class 9 Mathematics Chapter 6 Notes for students to gain conceptual clarity of the chapter. These notes are curated by subjectmatter experts according to the latest CBSE Syllabus. These notes will not only help them prepare for the Class 9 exam, but also in the preparation for competitive exams like IIT, JEE, Olympiads, and more.
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Access Class 9 Mathematics Chapter 6 – Lines and Angles Notes
Geometrical Concepts
Point:
A point is a precise position. It is a small dot that has only a location and no length, width, or thickness, i.e. no magnitude. It is denoted by capital letters A, D, C, O, etc.
Line Segment:
It is a straight path that connects two points and has defined endpoints and lengths. It has no width or thickness.
Ray:
A line segment that can only be extended in one direction is a ray.
Line:
It is formed when a line segment is extended in both directions indefinitely.
Collinear Points:
Two or more points that are on the same line are called collinear points.
Noncollinear Points:
The points that do not lie on the same line are noncollinear points.
Intersecting Lines:
Lines that intersect at a point are referred to as intersecting lines. Their common point is referred to as the point of intersection.
Concurrent Lines:
They are two or more lines intersecting at the same point.
Plane:
A surface on which every point of a line connecting any two points lies on that line is a plane. Its examples are the surface of a smooth wall, the surface of paper etc.
Angles:
When two straight lines intersect at a point, an angle is formed.
 An angle can be represented as ∠AOB or
AOB
 The arms of ∠AOB are OA and OB.
 The vertex of the angle (O) is the place where the arms meet.
 The amount of turning from one arm OA to other OB is known as the measure of the angle.
 ∠AOB can be written as m∠AOB.
 To calculate an angle, degrees, minutes and seconds are used.
 A ray is said to have turned 360° when it spins in an anticlockwise direction around its initial position and returns to its original position after one complete rotation.
Each rotation is divided into 360 equal parts and each part is 1°.
Each part (1∘ is divided into 60 equal parts, each of which is one minute (1′) long.
Each of the 60 equally sized portions of 1′ measures 1 second (1′′).
Degrees → Minutes → Seconds
1∘=60′
1′=60′′
By keeping in mind that the intersection of two rays creates an angle, we can infer this by looking at the various types of angles in the graphic below.
 ∠AOB – Acute angle (0∘ ∠AOB 90∘)
 ∠AOC – Right angle (=90∘)
 ∠AOD – Obtuse angle
(90∘ ∠AOD 180∘)
 ∠AOE – Straight angle (=180∘)
 ∠AOF Reflex angle, measured anticlockwise (180∘
∠AOF 360∘)
Acute Angle:
An angle whose measure is less than one right angle (that is, less than 90∘) is an acute angle.
Right Angle:
A right angle is one whose measure is 90∘.
Obtuse Angle:
An obtuse angle has a measure greater than one right angle and less than two right angles, or less than 180 degrees and greater than 90 degrees.
Straight Angle:
An angle whose measure is 180∘ is known as a straight angle.
Reflex Angle:
A reflex angle is an angle whose measure is more than 180∘ and less than 360∘. A reflex angle can be written as ref. ∠AOB
Complete Angle:
A complete angle is one whose angle is 360∘.
Equal Angles:
Two angles are said to be equal when two angles have the same measure.
Adjacent Angles:
Two angles sharing a common vertex and a common arm and having their other arms on opposite sides of the common arm are known as adjacent angles.
Complementary Angles:
They are those angles in which the total of the two angles is one right angle (that is 90∘).
Supplementary Angles:
If the sum of the two angles’ measurements is 180∘, they are referred to as supplementary angles.
Vertically Opposite Angles:
They are formed when two straight lines form pairs of opposite angles by intersecting each other at a point.
Bisector of an Angle
 If a ray or a straight line passing through the vertex of an angle divides the angle into two equal angles, it is the bisector of that angle.
∠BOC = ∠COA and
∠BOC + ∠COA= ∠AOB and.
∠AOB=2∠AOC=2C∠COA
Parallel Lines
 Two lines are parallel if they are coplanar and do not overlap even when they are extended on each side.
 However, there are lines that don’t intersect but aren’t parallel.
Transversal
A line that at different points intersects (or slices) two or more parallel lines is a transversal.
Angles Formed by a Transversal
Due to their placements, some of the angles can be grouped. Special names are given to such paired angles (apart from vertical angles and adjacent angles).
Interior angles which are on the same side of the Transversal ∠AQR (∠4) and ∠QRC (∠5), ∠BQR (∠3) and ∠QRD (∠6) form two pairs of interior angles on the same side of the transversal. Thus, interior angles can form on the same side of the transversal.
Alternate Angles
A pair of angles are called alternate angles if both of them are internal angles, they’re on opposing sides of the transversal axis, and are not adjacent angles. Alternate interior angles is another term for alternate angles.
Corresponding Angles
A pair of angles are called corresponding angles if one is an exterior angle and the other is an interior angle. They are not adjacent angles and are in the same transverse plane.
Parallel Lines – Theorem 1
 Statement:
When a transversal intersects two parallel lines, each pair of alternating angles is equal.
 Given:
ΔABC, side BC is produced to D and ∠ACD is the exterior angle formed and EFGH is a transversal.
 To Prove:
∠AFD=∠FGD (One pair of interior alternate angles) and ∠BFG=∠FGC (Another pair of interior alternate angles)
 Proof:
∠AFG= ∠EFB (Vertically opposite angles)
But ∠EFB=∠FGD (Corresponding angles)
∴∠AFG=∠FGD
Now,
∠BFG + ∠AFG=180∘…..(i)(Linear Pair)
∠FGC + ∠FGD=180∘……(ii)(Linear Pair)
From (i) and (ii)
∠BFG + ∠AFG=∠FGC + ∠FGD
But ∠AFG=∠FGD (Hence proved)
and ∠BFG=∠FGC
The converse of Theorem 1
 Statement:
If a pair of alternate interior angles are equal when a transversal intersects two lines, the two lines are parallel.
 Given:
Transversal EFGH intersects lines AB and CD such that a pair of alternate angles are equal. (∠AFD=∠FGD)
 To Prove:
ABCD
 Proof:
∠AFG=∠FGD (Given)
But ∠AFG=∠EFB(Vertically opposite angles)
∴∠EFB=∠FGD (Corresponding angles)
Therefore, (Corresponding angles axiom)
Parallel Lines – Theorem 2
 Statement:
Each set of consecutive interior angles is supplementary or additional when two parallel lines are connected by a transversal.
 Given:
ABCD and EFGH is a transversal.
 To Prove: ∠BFG + ∠FGD=180°
∠AFG + ∠FGC=180°
 Proof:
∠EFB + ∠BFG=180° (Linear Pair)
But ∠EFB=∠FGD (Corresponding angles axiom)
∴BFˆG+FGˆD=180°(Substitute FGˆD for EFˆB)
Similarly, we can prove that
∠AFG + ∠FGC=180°
The converse of Theorem 2
 Statement:
The two lines are parallel if a transversal intersects two lines in such a way that a pair of consecutive interior angles are supplementary.
 Given:
Transversal EFGH intersects lines AB and CD at F and G such that ∠BFG and ∠FGD are supplementary. That is (∠BFG + ∠FGD =180°)
 To Prove:
ABCD
 Proof:
∠EFB + ∠BFG=180∘……(i) (Linear pair (ray FB stands on EFGH))
(Corresponding angles postulate)
∠BFG + ∠FGD=180∘…..(ii) (Given)
∠EFB + ∠BFG=∠BFG + ∠FGD
Therefore, ∠EFB=∠FGD
(Subtract ∠BFG from both sides)
Since these are corresponding angles,
Therefore,
ABCD
Interior and Exterior Angles of a Triangle
An angle in a triangle refers to the angle formed by the two sides. The three angles located in the interior of the triangle are known as the triangle’s inner angles. Apart from this, there can be three external angles since a triangle has three sides. The interior angles opposite to the vertices where the exterior angles are formed are known as the interior opposite angles.
Triangles – Theorem
 Statement:
The exterior angle formed when a side of a triangle is produced is equal to the sum of the interior opposite angles.
 Given:
In triangle ΔABC, side BC is produced to D and ∠ACD is the exterior angle formed. ∠ABC and ∠BAC are the interior opposite angles.
 To Prove:
∠ACD=∠ABC+∠BAC
 Proof:
∠ABC + ∠BAC + ∠ACB
∠ABC + ∠BAC + ∠ACB =180∘……(i) (Theorem)
∠ACB + ∠ACD= 180∘…… (ii) (Linear pair)
From (i) and (ii), we get
∠ABC + ∠BAC + ∠ACB= ∠ACB + ∠ACD
Subtract ∠ACB from both the sides and we get: ∠ABC + ∠BAC= ∠ACD
Angle Sum Property
Three line segments unite three noncollinear points to make a plane closed geometric figure known as a triangle.
Triangles – Theorem 2:
 Statement:
The sum of three angles in a triangle is 180
 Given:
A triangle MNS
 To Prove:
∠M+∠N+∠S= 180
 Construction:
Draw a line AB parallel to the base NS using a scale through the vertex M.
 Proof:
NSAB
MN is a transversal
Therefore,
∠AMN=∠MNS…(1) Alternate angles
Similarly,
ABNS and MN
Therefore,
∠BMS=∠MSN…(2) Alternate angles
From the figure,
∠AMN + ∠NMS + ∠BMS= 180
Since, AB is a straight line and sum of angles at M =180
From (1) and (2)
∠MNS + ∠NMS + ∠MSN=180° by substituting ∠MNS and ∠MSN.
Thus, it is established that the sum of the angles’ three measurements equals 180, or two right angles.
Lines and Angles Class 9 Notes NCERT
Lines and Angles Notes Class 9
Class 9 Chapter 6 Mathematics Notes are easily accessible on Extramarks. Students can study from these notes at their own comfort and pace. They do not need any internet connection to refer to these notes. They can neatly arrange these Chapter 6 Mathematics Class 9 Notes in a folder on their laptops or mobiles and refer to them when in doubt. Students can even refer to these notes before any competitive exams for their preparation.
NCERT Class 9 Mathematics Chapter 6 Notes Revision
Class 9 Mathematics Notes Chapter 6 provided by Extramarks covers all topics of the chapter in detail. Here are some key important points of this chapter:
 A dot that does not have any component is referred to as a point.
 When two different points are joined, a line is formed. There are no endpoints in it, and thus, it can be extended to infinity.
 A line that has two endpoints is referred to as a line segment.
 A line that has an endpoint at one end, but the other end stretches to infinity is referred to as a ray.
 Points that are on the same line are referred to as collinear points. Points that do not lie on the same line are referred to as noncollinear points.
Angles
When two rays start from the same endpoint, an angle is made and the two rays form the angle’s arms. The endpoints are known as the vertex of the angle.
The Types of Angles
An angle that is between 0 and 90 degrees is known as an acute angle.
An angle that is equal to 90 degrees is known as a right angle.
An angle that lies between 90 and 180 degrees is known as an obtuse angle.
An angle that lies between 180 and 360 degrees is known as a reflex angle.
An angle that is equal to 180 degrees is known as a straight angle.
An angle that is equal to 360 degrees is known as a complete angle.
Complementary and Supplementary Angles
If the sum of the angles is equal to 90 degrees, then they are referred to as complementary angles.
If the sum of the angles is equal to 180 degrees, then they are referred to as supplementary angles.
Angles that have a similar vertex and one of their arms is common are referred to as adjacent angles.
Two angles that have the same vertex and one common arm are known as linear pairs of angles. The arms of the angles that are not common make a line.
Where two lines intersect each other at one point, then the opposite angle is known as the vertically opposite angle.
Intersecting and NonIntersecting Lines
Those lines that cross each other from one particular point are intersecting lines while nonintersecting lines are the ones that never cross each other at one point. Nonintersecting lines are parallel lines and the common distance between these lines always stays the same.
The Pairs of Angles Axioms
The sum of its two adjacent angles formed by the ray is 180 degrees if the rays stand on one line.
The arm that is not common to the angle forms a line if the sum of the two adjacent angles is 180 degrees.
FAQs (Frequently Asked Questions)
1. Explain the theorem of vertically opposite angles.
The theorem of vertically opposite angles explains that when two lines intersect each other, then the vertically opposite angles formed are always equal.
2. Define a transversal line.
If a line passes through two lines that are distinct and the line intersects the lines at distance points, then the line is said to be a transversal line.
3. What are collinear and noncollinear points?
If two points lie on the same line, they are collinear points. The points that do not lie on the same line are known as noncollinear points.
4. Explain the different types of angles.
Angles are of the following types:
 An angle that is between 0 and 90 degrees is known as an acute angle.
 An angle that is equal to 90 degrees is known as a right angle.
 An angle that lies between 90 and 180 degrees is known as an obtuse angle.
 An angle that lies between 180 and 360 degrees is known as a reflex angle.
 If the angle is equal to 180 degrees, it is a straight angle.
 A complete angle is equal to precisely 360 degrees.
5. What are intersecting and nonintersecting lines?
Those lines that cross each other from one particular point are intersecting lines while nonintersecting lines are the ones that never cross each other at one point. Nonintersecting lines are parallel lines and the common distance between these lines always stays the same.