**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-**

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 1-** If a ray stands on a line, then the sum of two adjacent angles so formed is 180^{0}.

**Given –** Line AB and a ray OC.

**Construction –** Draw a perpendicular DO at point O.

**(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.

**Helping Topics**