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Home » Lines and Angles » Parallel Lines » Two Lines Parallel to a Third are Parallel to Each Other

Two Lines Parallel to a Third are Parallel to Each Other

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will show that two lines parallel to a third line are parallel to each other.

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.

Geometry drawing: two lines parallel to a third line

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

« Tangent Line to a Circle
Parallelogram Formed by Connecting the Midpoints of a Quadrilateral »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

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    • Triangle Inequalities

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