• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
Geometry Help
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
menu icon
go to homepage
search icon
Homepage link
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
×
Home » Lines and Angles » Perpendicular lines » Converse Perpendicular Transversal Theorem

Converse 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 learn a step-by-step proof of the Converse Perpendicular Transversal Theorem: If two lines are perpendicular to a 3rd line, then they are parallel to each other.

In the previous problem, we showed that if a transversal line is perpendicular to one of two parallel lines, it is also perpendicular to the other parallel line.

Now we will show the opposite - that if a transversal line is perpendicular to two lines, then those two lines are parallel.

Prove:

If two lines are both perpendicular to one other line, then they are parallel to each other.

Given:

Converse Perpendicular Transversal Theorem proof: Line perpendicular to 2 parallel lines

L3 ⊥  L1, L3 ⊥  L2, prove L1 || L2

Proof

(1)    L3 ⊥  L1                                  //given

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

(3)    L3 ⊥  L2                                  //given

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

(5)    m∠1 = m∠2                         // transitive property of equality, both equal 90°

(6)    ∠1 ≅ ∠2                              //from (5) and definition of congruency

L1 || L2                                  //two lines are parallel if the corresponding angles formed by a transversal line are congruent

Strategy for this problem:

To show that two lines are parallel, we typically need to find two corresponding angles that are equal. The corresponding angles here are ∠1 ND ∠2, and using the facts given in the problem - that these are both right angles (since both L1 and L2 lines are perpendicular to L3), they are equal.

And that's how we prove the Converse Perpendicular Transversal Theorem.

Perpendicular Transversal 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]

Primary Sidebar

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

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy.


Copyright © 2023