Parallel Lines and Transversals

Key Concepts

• Define a transversal.
• Study about all the angles formed when a transversal intersects two parallel lines.
• Learn the Congruent angles postulate, Alternate interior angle theorem, Alternate exterior angle theorem, Consecutive interior angles theorem.
• Learn the Converse of Congruent angles postulate, Converse of Alternate interior angle theorem, Converse of Alternate exterior angle theorem, Converse of Consecutive interior angles theorem.

Parallel Lines and Transversals

Which geometric term would you use to describe the edges of sidewalks shown? 

The edges of sidewalks do not meet at any point and are at equal distance from each other throughout. So, the edges are parallel. 

Now, let us suppose that the boy reaches the finish line 

It looks like a line intersecting two parallel lines. 

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

Angles formed by a transversal 

• When a transversal passes through two lines, the angles that have the corresponding positions are called corresponding angles. 

∠1 and ∠2 are corresponding angles. 

• When a transversal passes through two lines, the angles that lie between the lines and on the opposite sides of the transversal are called as alternate interior angles. 

∠1 and ∠2 are alternate interior angles. 

• When a transversal passes through two lines, the angles that lie outside the two lines but on the opposite sides of the transversal are called as alternate exterior angles. 

∠1 and ∠2 are alternate exterior angles. 

• When a transversal passes through two lines, the angles that lie between the parallel lines on the same sides of the transversal are called as consecutive interior angles. 

∠1 and ∠2 are consecutive interior angles

Theorems/ Postulates 

Corresponding angles postulate

When two parallel lines are cut by a transversal then the pairs of corresponding angles are congruent. 

If m∥n and transversal t cuts the parallel line m and n, then ∠1=∠2

Alternate interior angles theorem 

When two parallel lines are cut by a transversal then the pairs of alternate interior angles are congruent. 

If m∥n and transversal t cuts the parallel lines m and n, then ∠1=∠2

Alternate exterior angles theorem 

When two parallel lines are cut by a transversal then the pairs of alternate exterior angles are congruent. 

If m∥n and transversal t cuts the parallel lines m and n, then ∠1=∠2

Consecutive interior angles theorem 

When two parallel lines are cut by a transversal then the pairs of consecutive interior angles are supplementary. 

If m∥n and transversal t cuts the parallel lines m and n, then ∠1 and ∠2 are supplementary. 

Transitive property of parallel lines 

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

Converse of theorems/ Postulates 

With the angles formed by a transversal, we can prove if the lines are parallel. 

We can learn  

1. Converse of corresponding angles postulate 
2. Converse of alternate interior angles theorem 
3. Converse of alternate exterior angles theorem 
4. Converse of consecutive interior angles theorem 

Converse of Corresponding angles postulate

“If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.” 

Converse of Alternate interior angles postulate

“If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.” 

Converse of Alternate exterior angles theorem

“If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.” 

Converse of consecutive interior angles theorem

“If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.” 

Exercise

1. Find the measure of a and b

2. Find the measure of the unknown angles x and y.

3. Can you prove that lines a and b are parallel? If so, explain how.

4. Use the diagram. Which rays are parallel? Which rays are not parallel? Justify your conclusions.

Concept Map

What we have learnt

• First, we have discussed about Parallel Lines and Transversals.
• A transversal is a line that intersects two or more coplanar lines at different points.
• The corresponding angles, alternate interior angles, alternate exterior angles formed by a transversal when it intersects a pair of parallel lines are congruent.
• The consecutive interior angles formed by a transversal when it intersects a pair of parallel lines are supplementary.

FAQs 

1. What is corresponding angles? 

Ans) The angles that have the corresponding positions when a transversal passes through two lines are known as corresponding angles. 

2. What is Alternate interior angles theorem? 

Ans) The pairs of alternate interior angles are congruent when a transversal cuts two parallel lines. 

3. What is Consecutive interior angles theorem? 

Ans) When a transversal cuts two parallel lines, the pairs of consecutive interior angles are supplementary. 

4. What is the relation between Parallel Lines and Transversals? 

Ans) Each pair of internal angles on the same side of a transversal is supplemental if the transversal intersects two parallel lines.