In geometry, parallel lines are lines that do not meet. They do not intersect or touch each other at any point. This definition first appeared very early, in Book I of Euclid's Elements.

One way to show that two lines are parallel to each other is to find two corresponding angles created by a transversal line that are congruent, as we have proven in this lesson.

We can use this theorem to come up with another way to show two lines are parallel- if they are both parallel to a third line.

## Problem

If line a is parallel to b, and line c is also parallel to b, show that line a is parallel to c.

## Strategy

We can show lines are parallel by finding a pair of corresponding angles that are congruent. Parallel lines form congruent angles when intersected by a third line. So let's draw a transversal line that intersects all three lines.

Line a is parallel to b, so ∠1 ≅ ∠2 which means that m∠1 = m∠2. Similarly, c is also parallel to b, so ∠2 ≅ ∠3 which means that m∠2 = m∠3. Using the transitive property of equality, m∠1 = m∠3, and so ∠1 ≅ ∠3.

And now, by using the converse of the corresponding angles theorem, line a is parallel to line c.

## Proof

This is how you show that two lines parallel to a third are parallel to each other:

(1) a||b //Given

(2) ∠1 ≅ ∠2 //corresponding angles formed by third line intersecting parallel lines are equal

(3) m∠1 = m∠2 //Definition of congruent angles

(4) b||c //Given

(5) ∠2 ≅ ∠3 //corresponding angles formed by third line intersecting parallel lines are equal

(6) m∠2 = m∠3 //Definition of congruent angles

(7) m∠1 = m∠3 //(3), (6), transitive property of equality

(8) ∠1 ≅ ∠3 //(7), definition of congruent angles

(9) a||c // Converse of the Corresponding Angles Theorem