# Parallel Lines and a Transversal

## Parallel lines with transversal

A **line** is a breadthless length which has no end point.

Here, AB is a line and it is denoted by AB.

A **line segment** is a part of a line which has two end points.

Here, AB is a line segment and it is denoted by AB.

A **ray** is a part of a line which has only one end point.

Here, AB is a ray and it is denoted by AB.

Three or more points which lie on the same line are called **collinear points**.

Three or more points which do not lie on a straight line are called **non-collinear points**.

**Parallel lines with a transversal**

- 1=5,2=6,4=8 and 3=7(Corresponding angles)
- 3=5,4=6 (Alternate interior angles)
- 1=7,2=8 (Alternate exterior angles)

**Lines parallel to the same line**

Lines that are parallel to the same line are also parallel to each other.

# Introduction to Geometry

## Angles and types of angles

When 2 rays originate from the same point at different directions, they form an angle.

An **angle** is formed when two rays originate from the same end point. The rays making an angle are called the **arms** of the angle. The end point from the two rays forming the angle originate is called the **vertex** of the angle.

- The rays are called arms and the common point is called vertex
**Types of angles :**- Acute angle 0<a<90
- Right angle a=90
- Obtuse angle : 90<a<180
- Straight angle =180
- Reflex Angle 180<a<360
- Angles that add up to 90 are complementary angles
- Angles that add up to 180 are called supplementary angles.

**Types of angles**:

Two angles whose sum is 90° are called **complementary angles.**

Two angles whose sum is 180° are called **supplementary angles.**

# Intersecting Lines and Associated Angles

## Intersecting and Non-Intersecting lines

When 2 lines meet at a point they are called **intersecting**

When 2 lines never meet at a point, they are called **non-intersecting** or parallel lines

**Adjacent angles**

*2 angles are adjacent if they have the same vertex and one common point.*

### Adjacent angles

Two angles are **adjacent**, if they have a common vertex, a common arm and their non–common arms are on different sides of the common arm.

▢*ABD* and ▢*DBC *are adjacent angles.

If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice-versa. This property is called as the **linear pair axiom**.

▢*ABD* and ▢*DBC* are linear pair of angles and ▢*ABD* + ▢*DBC* = 180^{o}.

## Linear Pair

When 2 adjacent angles are supplementary, i.e they form a straight line (add up to 180∘), they are called a linear pair.

**Vertically opposite angles**

When two lines intersect at a point, they form equal angles that are vertically opposite to each other.

# Basic Properties of a Triangle

**Triangle and sum of its internal angles**

Sum of all angles of a triangle add up to 180∘

**Exterior angle of a triangle = sum of opposite internal angles**

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

**∠4 is the exterior angle – ∠4=∠1+∠2**

If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.

The **vertically opposite angles** formed when two lines intersect each other. There are two pairs of vertically opposite angles.

▢*AOD* and ▢*BOC* , ▢*AOC* and ▢*BOD* are pair of vertically opposite angles.

If two lines intersect each other, then the **vertically opposite angles are equal**.

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

**Pair of angles when a transversal intersects two lines**:

**Interior angles on the same side of the transversal/ co-interior angles/ allied angles/ consecutive interior angles**:

**(i)** ▢4 and ▢5

**(ii)** ▢3 and ▢6

If a transversal intersects two parallel lines, then Each pair of **corresponding angles is equal**.

Each pair of **alternate interior angles is equal**.

Each pair of **interior angles on the same side of the transversal is supplementary**.

If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.

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

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.

Lines which are parallel to the same line are parallel to each other.

The sum of the angles of a triangle is 180°. This is known as the **angle sum property** **of a triangle**.

If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. This is known as the **exterior angle property of a triangle**.

An exterior angle of a triangle is greater than either of its interior opposite angles.