What are parallel lines? | Properties of Transversal of Parallel LlINES

A topic on Transversal of lines and Parallel lines has been described on the website link

In this topic we will learn the properties of Transversal of Parallel Lines and its converse.

Let us consider two Parallel lines viz., line **p** and line **q** (**line p ****∥ line q**) intersected by a transversal line **m**.

In the above figure we have, **line p ****∥ line q **and line **m** is the transversal of these two parallel lines. It is established that, if a transversal intersect two parallel Lines then

1. Pairs of Corresponding angles are equal (It is an Axiom called Corresponding Angles Axiom and does not require any proof to establish)

**∠1 = ∠****5**

**∠2 = ∠6**

**∠3 = ∠7**

**∠4 = ∠8**

**2. **Pairs of Alternate Interior angles are equal.

**∠3 = ∠6**

**∠4 = ∠5**

** 3. **Pairs of Alternate Exterior Angles are equal.

**∠1 = ∠8**

**∠2 = ∠7**

** **

4. Pairs of Interior Angles on the same side of Transversal are Supplementary that the sum of Interior Angles on the same side of the Transversal is 180°

**∠4 + ∠6 = ****180° **and **∠3 + ∠5 = ****180°**

** ****Sufficient Conditions for Parallelism of Two Lines:**

Two lines intersected by a Transversal are parallel to each other if and only if

- Any one pair of corresponding sides are equal. ( It is the converse of Corresponding Angles Axiom and does not require any proof to establish)
- Any one pair of Alternate Interior Angles are equal.
- Any one Pair of Alternate Exterior Angles are equal.
- Any one pair of Interior angles on the same side of the transversal are supplementary that is, the sum of any one pair of interior angles on the same side of transversal is 180°

Exercise:

Prove that if a Transversal Intersect 2(two) Parallel LlINES, then Pairs of Alternate Interior angles are equal.

Proof:-

Let us consider two Parallel line viz., line **p** and line **q** (**line p ****∥ line q**) intersected by a transversal line **m**.

We have,

**line p ****∥ line q** (given in the statement)

So,

** **

Hence, it is proved that that if a Transversal Intersect 2(two) Parallel Lines, then Pairs of Alternate Interior angles are equal.

__Self Assignment for the Students__

Prove that if a Transversal Intersect 2(two) Parallel Lines, then

- Pairs of Alternate Exterior Angles are equal.
- Pairs of Interior Angles on the same side of Transversal are supplementary.

** Note**: A brief description of Supplementary Angles has been made available on the website link

**Supplementrary Angles**

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