Lines and Angles Class 9 Notes CBSE Maths Chapter 6 (Free PDF Download)

Geometrical Concepts

Point:

• It is a precise position. 

• It is a small dot with no length, width, or thickness, but it does have location, i.e. no magnitude. 

• It is denoted by capital letters \[A,\text{ }B,\text{ }C,\text{ }O\] etc.

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Line Segment:

• A line segment \[\overline{AB}\] is a straight path that connects two points \[A\]and \[B\]. 

• It has a defined length and end points. (There is no width or thickness)

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Ray:

A ray is a line segment that can only be extended in one direction.

Line:

A line is formed when a line segment is stretched in both directions indefinitely.

Collinear Points:

Collinear points are defined as two or more points that are on the same line.

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Non-collinear Points:

• Non-collinear points are those that do not lie on the same line.

• Example: \[\text{A, B, C, D, E}\]

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Intersecting Lines:

• Intersecting lines are two lines that have a common point.

• The common point is called as the point of intersection.

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Concurrent Lines:

Concurrent lines are defined as two or more lines intersecting at the same point.

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Plane:

• A plane is a surface on which every point of a line connecting any two points lies on that line.

• Surface of a smooth wall, surface of a paper.

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Angles:

• An angle is formed when two straight lines intersect at a point.

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• It can be represented as \[\angle \text{AOB}\] or \[\text{A}\widehat{\text{O}}\text{B}\].

• \[\text{OA}\] and \[\text{OB}\]are the arms of \[\angle \text{AOB}\].

• The vertex of the angle \[\left( \text{O} \right)\] is defined as the place where the arms meet.

• The amount of turning from one arm \[\text{OA}\] to other \[\text{OB}\] is called the measure of the angle \[\angle \text{AOB}\] and written as \[\text{m}\angle \text{AOB}\].

• Degrees, minutes, and seconds are used to calculate an angle.

• If a ray spins in an anticlockwise direction around its initial position and returns to its original position after one complete rotation, it has turned \[{{360}^{\circ }}\].

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\[1\] complete rotation is divided into \[360\] equal parts. 

Each part is \[{{1}^{{}^\circ }}\]. 

Each part \[\left( {{1}^{\circ }} \right)\] is divided into \[60\] equal parts where, each part measures \[1\] minute \[\left( 1' \right)\]. 

\[1'\] is divided into \[60\] equal parts where, each part measures \[1\] second \[\left( 1'' \right)\]

\[\text{Degrees }\to \text{ minutes }\to \text{ seconds}\]

\[{{1}^{\circ }}=60'\]

\[1'=60''\]

By recalling that the union of two rays forms an angle.

By observing the different type of angles in below figure, we conclude that

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• \[\text{A}\widehat{\text{O}}\text{B}\] is an acute angle \[\left( {{\text{0}}^{\circ }}\text{  A}\widehat{\text{O}}\text{B  9}{{\text{0}}^{\circ }} \right)\]

• \[\text{A}\widehat{\text{O}}\text{C}\] is a right angle \[\left( \text{an angle equal to 9}{{\text{0}}^{\circ }} \right)\]

• \[\text{A}\widehat{\text{O}}\text{D}\] is an obtuse angle \[\left( \text{9}{{\text{0}}^{\circ }}\text{  A}\widehat{\text{O}}\text{D  18}{{\text{0}}^{\circ }} \right)\]

• \[\text{A}\widehat{\text{O}}\text{E}\] is a straight angle \[\left( \text{an angle equal to 18}{{\text{0}}^{\circ }} \right)\]

• \[\text{A}\widehat{\text{O}}\text{F}\] (measured in anti-clock wise direction) is a reflex angle \[\left( \text{18}{{\text{0}}^{\circ }}\text{  A}\widehat{\text{O}}\text{F  36}{{\text{0}}^{\circ }} \right)\]

Right Angle:

An angle whose measure is \[\text{9}{{\text{0}}^{\circ }}\] known as a right angle.

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Acute Angle:

An angle whose measure is less than one right angle (that is, less than  \[\text{9}{{\text{0}}^{\circ }}\]), known as an acute angle.

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Obtuse Angle:

An angle whose measure is more than one right angle and less than two right angles (that is less than \[\text{18}{{\text{0}}^{\circ }}\]and more than \[\text{9}{{\text{0}}^{\circ }}\]) known as an obtuse angle.

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Straight Angle:

An angle whose measure is \[\text{18}{{\text{0}}^{\circ }}\] known as a straight angle.

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Reflex Angle:

An angle whose measure is more than \[\text{18}{{\text{0}}^{\circ }}\]and less than \[\text{36}{{\text{0}}^{\circ }}\]is called a reflex angle.

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It can be written as \[\text{ref}\text{. }\angle \text{AOB}\].

Complete Angle:

An angle whose measure is \[\text{36}{{\text{0}}^{\circ }}\] called a complete angle. 

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Equal Angles:

When two angles have the same measure, they are said to be equal.

Adjacent Angles:

Adjacent angles are two angles that share a common vertex and a common arm and have their other arms on opposite sides of the common arm.

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\[O\] is the common vertex.

\[\text{A}\widehat{\text{O}}\text{B}\] and \[B\widehat{\text{O}}C\] are adjacent angles.

Arm \[\text{BO}\] separates the two angles.

Complementary Angles:

Complementary angles are those in which the total of the two angles is one right angle (that is \[\text{9}{{\text{0}}^{\circ }}\]).

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If the measure of 

\[\text{A}\widehat{\text{O}}C={{\text{a}}^{\circ }}\]

\[\text{C}\widehat{\text{O}}\text{B}={{b}^{\circ }}\], then

\[{{\text{a}}^{\circ }}+{{b}^{\circ }}={{90}^{\circ }}\]

Therefore, \[\text{A}\widehat{\text{O}}C\] and \[\text{C}\widehat{\text{O}}\text{B}\] are complementary angles.

\[\text{A}\widehat{\text{O}}C\] is complement of \[\text{C}\widehat{\text{O}}\text{B}\].

Supplementary Angles:

If the sum of two angles' measurements is \[\text{18}{{\text{0}}^{\circ }}\], they are said to be supplementary.

Example:

Angles measuring \[\text{13}{{\text{0}}^{\circ }}\]and \[\text{5}{{\text{0}}^{\circ }}\] are supplementary angles. 

Two supplementary angles are mutually beneficial.

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Vertically Opposite Angles:

Vertically opposing angles are generated when two straight lines intersects each other at a point and form pairs of opposite angles.

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Angles \[\angle 1\] and  \[\angle 3\], and angles \[\angle 2\] and \[\angle 4\] are vertically opposite angles.

Vertically opposite angles are always equal.

Bisector of an Angle

• A ray or a straight line passing through the vertex of an angle is known as the Bisector of that angle if it divides the angle into two equal-sized angles.

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• \[\text{B}\widehat{\text{O}}\text{C = C}\widehat{\text{O}}A\] and,

\[\text{B}\widehat{\text{O}}\text{C + C}\widehat{\text{O}}A=A\widehat{\text{O}}B\] and.

\[A\widehat{\text{O}}B=2\text{B}\widehat{\text{O}}\text{C}=2\text{C}\widehat{\text{O}}A\]

Parallel Lines

• Even if they are extended on each side, two lines are parallel if they are coplanar and do not overlap.

• There are, however, lines that don't intersect yet aren't parallel. 

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• They're skew lines, as the name implies. Lines that are not coplanar and do not intersect are referred to as skew lines. 

• The lines \[\text{AE}\] and \[\text{HG}\] are skewed.

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Transversal

• Observe the three lines '\[\text{l}\]', '\[\text{m}\]' and '\[\text{t}\]'.

• In the diagram '\[\text{l}\]' and '\[\text{m}\]' are two parallel lines. '\[\text{t}\]' intersects '\[\text{l}\]' at two distinct points '\[\text{A}\]' and '\[\text{B}\]' and '\[\text{m}\]' at '\[\text{C}\]' and '\[\text{D}\]'. Line t is known as transversal.

• A transversal is a line that at different points intersects (or slices) two or more parallel lines.

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Angles Formed by a Transversal 

• In the diagram \[\overleftrightarrow{\text{AB}}\] and \[\overleftrightarrow{\text{CD}}\] are two parallel lines. 

\[\text{PQRS}\] is a transversal intersecting \[\overleftrightarrow{\text{AB}}\] at \[Q\] and \[\overleftrightarrow{\text{CD}}\] at \[\text{R}\]. 

There are total eight angles formed.

• Some of the angles can be grouped together due to their placements. Special names are given to the paired angles (apart from adjacent angles and vertical angles).

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Interior Angles which are on the same side of the Transversal

• From the below figure,

\[A\widehat{Q}\text{R }\left( \angle 4 \right)\] and \[Q\widehat{R}\text{C }\left( \angle 5 \right)\] and \[B\widehat{Q}\text{R }\left( \angle 3 \right)\] and \[Q\widehat{R}\text{D }\left( \angle 6 \right)\] form two pairs of interior angles on the same side of the transversal.

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Alternate Angles

• A pair of angles are said to be alternate angles if

1. both angles are internal angles,

2. they're on opposing sides of the transversal axis, and

3. they are not adjacent angles, they are said to be alternate angles.

• Alternate interior angles are another name for alternate angles.

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• In the above figure,

\[A\widehat{Q}\text{R}\]and \[Q\widehat{R}\text{D}\] \[\left( \angle \text{4 and }\angle 6 \right)\]

\[B\widehat{Q}\text{R}\] and \[Q\widehat{R}\text{C}\] \[\left( \angle \text{3 and }\angle 5 \right)\] are the two pairs of alternate angles.

Corresponding Angles 

• A pair of angles are said to be corresponding angles if 

• One is an interior angle and the other is an exterior angle 

• They are in the same transverse plane and 

• They are not adjacent angles. 

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The four pairs of corresponding angles are given as follows;

\[A\widehat{Q}\text{P}\]and \[C\widehat{R}\text{Q}\] \[\left( \angle \text{1 and }\angle 5 \right)\]

\[A\widehat{Q}\text{R}\] and \[C\widehat{R}\text{E}\] \[\left( \angle \text{4 and }\angle 8 \right)\]

\[B\widehat{Q}\text{R}\] and \[D\widehat{R}\text{E}\] \[\left( \angle \text{3 and }\angle 7 \right)\]

Parallel Lines - Theorem 1

• Statement: 

Each pair of alternating angles is equal when a transversal intersects two parallel lines.

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• Given: 

$\Delta \text{ABC}$, side $\text{BC}$ is produced to $D$ and \[\text{A}\widehat{\text{C}}\text{D}\]is the exterior angle formed.

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\] and \[\text{EFGH}\] is a transversal.

• To Prove: 

\[\text{A}\widehat{F}D=F\widehat{G}D\] (One pair of interior alternate angles)

\[B\widehat{F}G=F\widehat{G}C\] (Another pair of interior alternate angles)

• Proof:

\[\text{A}\widehat{F}G=E\widehat{F}B\] (Vertically opposite angles)

But

\[E\widehat{F}B=F\widehat{G}D\] (Corresponding angles)

\[\therefore \text{A}\widehat{F}G=F\widehat{G}D\]

Now,

\[B\widehat{F}G+\text{A}\widehat{F}G={{180}^{\circ }}.....\left( \text{i} \right)\left( \text{Linear Pair} \right)\]

\[F\widehat{G}C+F\widehat{G}D={{180}^{\circ }}.....\left( \text{ii} \right)\left( \text{Linear Pair} \right)\]

From \[\left( \text{i} \right)\] and \[\left( \text{ii} \right)\],

\[B\widehat{F}G+\text{A}\widehat{F}G=F\widehat{G}C+F\widehat{G}D\]

But

\[\text{A}\widehat{F}G=F\widehat{G}\text{D}\left( \text{Proved} \right)\]

\[\therefore B\widehat{F}G=F\widehat{G}C\]

Converse of Theorem $1$

• Statement: 

If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel.

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• Given: 

Transversal \[\text{EFGH}\] intersects lines \[\text{AB}\] and \[\text{CD}\] such that a pair of alternate angles are equal.

\[\left( \text{A}\widehat{F}D=F\widehat{G}D \right)\]

• To Prove: 

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\]

• Proof:

\[\text{A}\widehat{F}G=F\widehat{G}D\] (Given)

But

\[A\widehat{F}G=E\widehat{F}B\]  (Vertically opposite angles)

\[\therefore E\widehat{F}B=F\widehat{G}D\] (Corresponding angles)

Therefore,

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\] (Corresponding angles axiom)

Parallel Lines - Theorem $2$

• Statement: 

Each set of consecutive interior angles is additional or supplementary when a transversal connects two parallel lines.

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• Given: 

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\] and \[\text{EFGH}\] is a transversal.

• To Prove: 

\[B\widehat{F}G+F\widehat{G}D={{180}^{\circ }}\]

\[A\widehat{F}G+F\widehat{G}C={{180}^{\circ }}\]

• Proof:

\[E\widehat{F}B+B\widehat{F}G={{180}^{\circ }}\] (Linear Pair)

But

\[E\widehat{F}B=F\widehat{G}D\] (Corresponding angles axiom)

\[\therefore B\widehat{F}G+F\widehat{G}D={{180}^{\circ }}\] (Substitute \[F\widehat{G}D\] for \[E\widehat{F}B\])

Similarly, we can prove that

\[A\widehat{F}G+F\widehat{G}C={{180}^{\circ }}\]

Converse of Theorem $2$

• Statement: 

If a transversal intersects two lines in such a way that a pair of consecutive interior angles are supplementary, then the two lines are parallel.

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• Given: 

Transversal \[\text{EFGH}\] intersects lines \[\text{AB}\] and \[\text{CD}\] at \[\text{F}\] and \[\text{G}\] such that \[B\widehat{F}G\] and \[F\widehat{G}D\] are supplementary.

That is \[\left( B\widehat{F}G+F\widehat{G}D={{180}^{\circ }} \right)\]

• To Prove: 

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\]

• Proof:

\[E\widehat{F}B+B\widehat{F}G={{180}^{\circ }}......\left( \text{i} \right)\] (Linear pair (ray \[\text{FB}\] stands on \[\text{EFGH}\]))

(Corresponding angles postulate)

\[B\widehat{F}G+F\widehat{G}D={{180}^{\circ }}.....\left( \text{ii} \right)\] (Given)

\[E\widehat{F}B+B\widehat{F}G=B\widehat{F}G+F\widehat{G}D\]

Therefore,

\[E\widehat{F}B=F\widehat{G}D\] (Subtract \[B\widehat{F}G\] from both sides)

Since these are corresponding angles,

Therefore,

\[\text{AB  }\!\!|\!\!\text{  }\!\!|\!\!\text{  CD}\] 

Interior and Exterior Angles of a Triangle

• When we talk about an angle in a triangle, we're talking about the angle formed by the two sides. 

• The three angles are located in the triangle's interior. These angles are known as the triangle's inner angles. 

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• Now look at the triangle that the sides are formed in.

• In the fig, is extended to $\text{O}$. An angle $\text{OXZ}$ is formed.

$\overline{\text{XZ}}$ is produced to $\text{P}$ forming an angle $\text{YZP}$. 

• Similarly, $\overline{\text{ZY}}$ is formed to $\text{M}$ forming angle $\text{MYX}$. 

• These angles $\text{OXZ, YZP}$and $\text{MYX}$X are called exterior angles of $\text{ABC}$ 

• There can be three external angles because the triangle has three sides. 

• The interior angles opposite to the vertices where the exterior angles are formed, are called the interior opposite angles. 

• From the figure, at $\text{X}$, the exterior angle $\text{OXZ}$ is formed and angles $\text{XYZ}$ and $\text{XZY}$are interior opposite angles of it.

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From above figure,

For exterior angle $1$; the interior opposite angles are \[\text{B}\widehat{A}C\] and \[\text{A}\widehat{B}C\].

For exterior angle $2$; the interior opposite angles are \[\text{A}\widehat{B}C\] and \[\text{A}\widehat{C}B\].

For exterior angle $3$; the interior opposite angles are \[\text{B}\widehat{A}C\] and \[\text{A}\widehat{C}B\].

Triangles - Theorem $1$:

• Statement: 

If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.

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• Given: 

In triangle, $\Delta \text{ABC}$, side $\text{BC}$ is produced to $D$ and \[\text{A}\widehat{\text{C}}\text{D}\]is the exterior angle formed.

\[\text{A}\widehat{B}C\] and \[B\widehat{A}C\] are the interior opposite angles.

• To Prove: 

\[\text{A}\widehat{C}D=\text{A}\widehat{B}C+\text{B}\widehat{A}C\]

• Proof:

\[A\widehat{B}C+B\widehat{A}C+A\widehat{C}B={{180}^{\circ }}......\left( \text{i} \right)\] (Theorem) 

\[A\widehat{C}B+A\widehat{C}D={{180}^{\circ }}......\left( \text{ii} \right)\] (Linear pair) 

From \[\left( \text{i} \right)\] and \[\left( \text{ii} \right)\], we get

\[\text{A}\widehat{B}C+\text{B}\widehat{A}C+\text{A}\widehat{C}B=\text{A}\widehat{C}B+\text{A}\widehat{C}D\]

Subtract \[\text{A}\widehat{C}B\] from both sides, we get

\[\text{A}\widehat{B}C+\text{B}\widehat{A}C=\text{A}\widehat{C}D\]

Angle Sum Property

Three line segments unite three non-collinear points to make a triangle, which is a plane closed geometric figure.

Triangles - Theorem $2$:

• Statement: 

The sum of the three angles of a triangle is ${{180}^{\circ }}$.

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• Given: 

A triangle $\text{MNS}$. 

• To Prove: 

$\widehat{M}+\widehat{N}+\widehat{S}={{180}^{\circ }}$

• Construction:

By using a scale through the vertex $\text{M}$, draw a line $\overleftrightarrow{AB}$ parallel to the base $\overline{NS}$.

• Proof:

$\overline{NS}||\overleftrightarrow{AB}$

$\text{MN}$ is a transversal.

Therefore,

$\text{A}\widehat{\text{M}}\text{N = M}\widehat{\text{N}}\text{S }.....\left( 1 \right)\text{ Alternate angles}$

Similarly, \[\overleftrightarrow{AB}||\overline{NS}\] and \[MN\]

Therefore,

$\text{B}\widehat{\text{M}}\text{S = M}\widehat{\text{S}}\text{N }.....\left( 2 \right)\text{ Alternate angles}$

From the figure,

$\text{A}\widehat{\text{M}}\text{N + N}\widehat{M}\text{S + B}\widehat{M}S={{180}^{\circ }}$

Since, \[\overleftrightarrow{AB}\] is a straight line and sum of the angles at $M={{180}^{\circ }}$

From $\left( 1 \right)$and $\left( 2 \right)$,

$M\widehat{N}\text{S + N}\widehat{M}\text{S + M}\widehat{S}N={{180}^{\circ }}$ - By substituting $M\widehat{N}\text{S}$ and $\text{M}\widehat{S}N$

Thus it is proved that sum of the measures of the three angles of a triangle is equal to ${{180}^{\circ }}$ or two right angles.

Lines and Angles Class 9 Notes NCERT

Lines and Angles Notes Class 9 PDF

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NCERT Class 9 Maths Chapter 6 Notes Revision

Our revision Class 9 Maths Ch 6 notes cover the topic in great detail. Here are the notes for this chapter.

• A point is a dot that does not have any component.

• A line is formed when two different points are joined. There are no endpoints in the line, and this can be extended till infinity.

• A line segment is a line that has two endpoints.

• A ray is a line that has an endpoint at one end, but the other end stretches to infinity.

• Points that are on the same line are collinear. Points that do not lie on the same line are non-collinear

Angles

An angle is made when two rays start from the same endpoint. The two rays form arms of the angle. The endpoints are the vertex of the angle. 

The Types of Angles

• An angle that is between 0 and 90 degrees is acute.

• An angle that is exactly equal to 90 degrees is a right angle

• An angle that lies between 90 and 180 degrees is obtuse.

• A reflex angle lies between 180 and 360 degree

• If the angle is exactly equal to 180 degrees, then this is a straight angle

• A complete angle is the one that is equal to precisely 360 degree

Complementary and Supplementary Angles

• If the sum of the angles is equal to 90 degrees then these are complementary angles

• If the sum of the angles is equal to 180 degrees, then this is supplementary angles

• Adjacent angles are angles that have a similar vertex, and one of their arms is common.

• Linear pairs of angles have two angles with the same vertex and have one common arm. The arms that are not common make a line.

• Where two lines intersect each other at one point, then the opposite angle is called the vertically opposite angle.

Intersecting and Non-Intersecting Lines

Intersecting lines are those lines that cross each other from one particular point. Non- intersecting lines are those that never cross each other at one point. These are parallel lines, and the common distance between the lines stays the same.

The Pairs of Angles Axioms

If the rays stand on one line, then the sum of its two adjacent angles formed by the ray is 180 degrees.

If the sum of the two adjacent angles is 180 degrees, then the arm that is not common to the angle forms a line.

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