**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 90^{o}, the angles are **complementary angles**.

**(ii)Supplementary angles.**

When the sum of the two angles is 180^{o}, the angles are **supplementary angles**.

**(iii) Adjacent 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 0^{0} and 90^{0}.

**(vii) Obtuse angle-**

An **obtuse angle** is an angle which measures greater than 90^{0} and less than 180^{0}.

**(viii) Right angle-**

A **right angle** is an angle which measures equal to 90^{0}.

**(ix) Straight angle**

A **straight angle** is an angle that measures equal to 180^{0}.

**(x)Reflex angle-**

A **reflex angle** is an angle which measures greater than 180^{0} but less than 360^{0}.

**(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 180^{0}.

**Explanation –**

∠AOC + ∠COB = 180^{0 }(Linear Pair Angles)

**Statement A (Converse of the Axiom 6.1)** – If sum of two adjacent angles is 180^{0}, 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 = 180^{0} (Given)

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

If the sum of two adjacent angles is 180^{0}, 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 = 180^{0}

And ∠AQR + ∠QRC = 180^{0}

**Proof –**

∠PQA = ∠QRC (Corresponding Angles)

And, ∠PQA + ∠AQR = 180^{0} (Linear Pair Angles)

∠QRC + ∠AQR = 180^{0} (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 = 180^{0}

And ∠AQR + ∠QRC = 180^{0}

**What to Prove –** AB is parallel to CD

**Proof –**

∠AQR + ∠QRC = 180^{0} (Given) …(1)

∠PQA + ∠AQR = 180^{0} (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 180^{0}.

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

**What to prove –** ∠A + ∠B + ∠C = 180^{0}

**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**