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Home » Lines and Angles » Perpendicular lines » Linear Pair Perpendicular Theorem

Linear Pair Perpendicular Theorem

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

This theorem states that if two straight lines intersect at a point, and the linear pair of angles they form have an equal measure, then the two lines are perpendicular to each other.

Problem

If two straight lines intersect at a point and form a linear pair of equal angles, they are perpendicular.

Two Right Angles

in other words, given m∠1 = m∠2, prove that the lines L1 and L2  are perpendicular.

Strategy

To prove that lines are perpendicular, we need to find an angle that measures 90°. In the problem statement, we are given the fact that we have a linear pair of angles, and that they are equal. The hint here is "linear pair" - we know those measure 180°, and 180° is twice 90°.

Proof

(1)    m∠1 = m∠2
(2)    m∠1 + m∠2 = 180°          // straight line measures 180° 
(3)    m∠1 + m∠1 = 180°           // using (2) and performing algebraic substitution, replacing  m∠2 with the equivalent m∠1
(4)    2 • m∠1 = 180°    // Simplify, using multiplication
(5)    m∠1 = 180°/2= 90°           //algebraically solve for m∠1
(6)    L1 ⊥  L2                                //per the definition of perpendicular lines.

« Converse of the Corresponding Angles Theorem
Lines and Angles »

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…

Geometry Topics

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