# Lines and Angles

Notes of chapter: Lines and Angles are presented below. Indepth notes along with worksheets and NCERT Solutions.

(1)Ray-

A line that has one end point and it extends infinitely in other direction, is called ray.

In the above diagram, PQ is a ray with end P.

(2) Line segment-

A part of a line that has two end points is called line segment.

In above diagram, PQ is a line segment with ends P and Q.

(3) Lne-

A line segment when extends its both end points in either direction endlessly, is called a line.

In the above diagram, PQ is a line with no ends.

(4) Vertex-

The meeting point or common point of two rays or two line segments or two lines is called vertex.

(5) Angle-

The distance between two rays or two line segments or two lines diverging from vertex (common point) is called an angle.

(6) Types of angles

(i) Complementary angles-

When the sum of the two angles is 90o, the angles are complementary angles.

(ii)Supplementary angles.

When the sum of the two angles is 180o, the angles are supplementary angles.

When two angles have a common vertex, a common arm and non – common arms are on either side of the common arm, are called adjacent angles.

(iv) Linear pair angles-

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.

(v)Vertically opposite angles

When two lines intersect, the vertically opposite angles so formed are equal.

(vi) Acute angle-

An acute angle is an angle which measures between 00 and 900.

(vii) Obtuse angle-

An obtuse angle is an angle which measures greater than 900 and less than 1800.

(viii) Right angle-

A right angle is an angle which measures equal to 900.

(ix) Straight angle

A straight angle is an angle that measures equal to 1800.

(x)Reflex angle-

A reflex angle is an angle which measures greater than 1800 but less than 3600.

(7)Pairs of Lines

(i)When two or more lines meet at a common point are called intersecting lines and this common point is called point of intersection.

In the above figure, l and m are intersecting lines and point O is point of intersection.

(ii)Transversal line-

A line that intersects two or more lines at distinct points is called a transversal line.

In the above figure, p is transversal to the lines l, m and n.

(iii) Angles made by a transversal

(a)Interior angles-

When two lines are cut by a third line (transversal), then the angles formed inside the lines are called interior angles.

(b)Exterior angle-

When two lines are cut by a third line (transversal), then the angles formed outside the lines are called exterior angle.

(c) Corresponding angles-

The angles which are on the same side of the transversal and have different vertices are called corresponding angles. They are in ’corresponding’ position (above or below, left or right) relative to the two lines.

(d) Alternate interior angles-

The angles which are on opposite sides of the transversal and have different vertices are called alternate interior angles. These angles lie ‘between’ the two lines.

(e) Alternate exterior angles-

The angles which are on opposite sides of the transversal and have different vertices are called alternate exterior angles. These angles lie outside the two lines.

(iv)Transversal of parallel lines

A line that intersects two or more parallel lines at distinct points is called a transversal of parallel line.

In the above figure, l is a transversal line and m & n are parallel lines.

Angles made by Transversal of Parallel lines

(a)If two parallel lines cut by a transversal, each pair of corresponding angles is equal in measure.

(b) If two parallel lines cut by a transversal, each pair of alternate interior angles is equal in measure.

(c) If two parallel lines cut by a transversal, each pair of interior angles on the same side of the transversal is supplementary.

(8)Checking for parallel lines

(i) When, a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel.

(ii) When, a transversal cuts two lines, such that pairs of alternate interior angles are equal, then the lines have to be parallel.

(iii) When, a transversal cuts two lines, such that pairs of interior angles on the same side of the transversal are supplementary, then the lines have to be parallel.

(9) Pairs of Angles

Axiom 6.1- If a ray stands on a line, then the sum of two adjacent angles so formed is 1800.

Explanation –

∠AOC + ∠COB = 1800 (Linear Pair Angles)

Statement A (Converse of the Axiom 6.1) – If sum of two adjacent angles is 1800, then a ray stands on a line (that is, the non – common arms from a line.)

In above figure,

AO and OB are non – common arms, which lie along a ruler(If we put a ruler along the non – common arms). It shows that points A, O and B lie on the same line.

Ray OC stands on the line AOB.

And, ∠AOC + ∠COB = 1800 (Given)

Therefore, we can state converse of axiom 6.1( axiom 6.2) as follows:

If the sum of two adjacent angles is 1800, then the non –common arms of the angles form a line.

The above axioms are called “Linear Pair Axiom”.

Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equals.

Given – Two lines AB and CD, which intersect each other at point O.

∠AOC, ∠COB, ∠BOD and ∠DOA are four angles that are made after lines intersect each other.

Axiom 6.3 – If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

Explanation –

∠EGA = ∠GHC (Corresponding angles)…..(1)

And

∠EGA= ∠BGH (Vertical opposite angles)…..(2)

From equation (1) and (2)

∠GHC = ∠BGH

Similarly

∠AGH = ∠GHD

Hence, proved.

Axiom 6.3 is also referred to as corresponding angles axiom.

Axiom 6.4 –  If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Construction-

Draw a Line AD and mark two points B and C on it.

Construct angle ∠ABE At point B.

Construct another angle ∠FCD at point C which is equal to ∠ABE.

Produce EB and FC on the other side of the line of AD to form lines EG and FH.

Construct a perpendicular from point L to line EG and draw another perpendicular from point M on line FH.

Explanation-

Line FC and EB do not touch each other.

Lengths of both perpendiculars are same and equal to BC.

Hence, line EB and FC are parallel lines.

Theorem  6.2– If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Given  – Two parallel lines AB and CD one transversal line PS.

What to prove – ∠AQR = ∠QRD and   ∠BQR = ∠QRC

Proof –

∠PQA = ∠QRC (Corresponding Angles Axiom) …(1)

∠PQB = ∠QRD (Corresponding Angles Axiom) …(2)

∠PQA = ∠BQR (Vertically Opposite Angles)

Put value of ∠PQA to equation (1)

∠BQR = ∠QRC (Alternate Interior Angles)

Similarly, we can prove that ∠AQR = ∠QRD (Alternate Interior Angles)

Hence, proved.

Theorem 6. 3– If a transversal intersects two lines such that a pair of alternate interior angle is equal, then the two lines are parallel.

Given – Alternate Angles ∠AQR = ∠QRD and alternate angles  ∠BQR = ∠QRC

What to prove – AB is parallel to CD

Proof –

∠BQR = ∠PQA (Vertically Opposite Angles)….(1)

∠BQR = ∠QRC (Given)….(2)

From equation 1 and 2

∠PQA = ∠QRC (Corresponding Angles)

Therefore, AB is parallel to CD (Converse of corresponding angles axiom)

Hence, proved.

Theorem  6.4– If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

Given – Two parallel lines AB and CD one transversal line PS.

What To Prove – ∠BQR + ∠QRD = 1800

And ∠AQR + ∠QRC = 1800

Proof –

∠PQA = ∠QRC (Corresponding Angles)

And, ∠PQA + ∠AQR = 1800 (Linear Pair Angles)

∠QRC + ∠AQR = 1800 (Supplementary Angles)

Hence proved.

Theorem 6.5 – If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Given – ∠BQR + ∠QRD = 1800

And ∠AQR + ∠QRC = 1800

What to Prove – AB is parallel to CD

Proof –

∠AQR + ∠QRC = 1800 (Given) …(1)

∠PQA + ∠AQR = 1800 (Linear Pair Angles)(2)

From equation 1 and 2

∠AQR + ∠QRC = ∠PQA + ∠AQR

∠QRC = ∠PQA (Corresponding Angles Axiom)

Therefore, AB is parallel to CD

Hence, proved.

(11) Lines parallel to the same line

If two lines are parallel to the same line, they are parallel to each other.

Given – Line m  line l and line n  line l.

Construction – Draw a transversal line t which intersects line l, m and n.

Proof –

∠1 = ∠2 and

∠1 = ∠3  (Corresponding angles axiom)

∠2 = ∠3  (Corresponding angles)

Therefore,

Line m is parallel to Line n

This result can be stated in the form of the following theorem:

Theorem  6.6 – Lines which are parallel to the same lines are parallel to each other.

Theorem  6.7- The sum of the angles of a triangle is 1800.

Given – A triangle ABC and angles A, B and C.

What to prove – ∠A + ∠B + ∠C = 1800

Construction – Draw a line XAY parallel to BC through the vertex A.

Theorem 6.8- If a side of triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Given – A triangle ABC and angles A, B and C.

What to prove – Exterior angle ACD = ∠CAB + ∠ABC

Construction – Side BC of triangle is produced to point C.

Exterior angle ACD formed.

Helping Topics

NCERT Solutions Class 7

NCERT Solutions Class 9

Worksheet Class 7

Worksheet Class 9