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Home » Lines and Angles » Perpendicular lines » Perpendicular Transversal Theorem

Perpendicular Transversal Theorem

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 see a step by step proof of the Perpendicular Transversal Theorem: if a line is perpendicular to 1 of 2 parallel lines, it's also perpendicular to the other.

In the section that deals with parallel lines, we talked about two parallel lines intersected by a third line, called a "transversal line".

The transversal line can intersect the parallel lines at any angle. When the angle of intersection is a right angle (90°), the transversal line is called a "perpendicular transversal", and we will now show that if it is perpendicular to one line, it is also perpendicular to the other parallel line.

Prove:

If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other one.

Given:

Perpendicular Transversal Theorem proof: Line perpendicular to 2 parallel lines

 L3 ⊥  L1, L1 || L2, prove L3 ⊥  L2

Proof

Here's how you prove the Perpendicular Transversal Theorem:

(1)  L3 ⊥  L1               // given

(2) m∠1 = 90°           // definition of perpendicular lines

(3) L1 || L2                 // given

(4)   m∠1 = m∠2        // corresponding angles of two parallel lines

(5)   m∠2 = 90°           //from (2) & (4) , using algebraic substitution

(6)   L3 ⊥  L2              // definition of perpendicular lines

Strategy for this problem:

Like almost any geometry problem involving parallel lines, we will prove this by finding corresponding angles. And, like any other proof problem, we will be making use of all the information given in the problem statement.

So, we were given the fact that L1 and L2 are parallel - so we use that and see the corresponding angles (∠1 and ∠2  ) - which we know are equal. We were also given the fact that m∠1 = 90°  since L3 and L1 are perpendicular lines. So, using both facts we have m∠1 = m∠2= 90°, and we are done.

« Converse Perpendicular Transversal Theorem
Vertical Angles Theorem »

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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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…

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