• 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 » Parallel Lines » Alternate Interior Angles Theorem

Alternate Interior Angles 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 prove the alternate interior theorem, stating that interior alternating angles and exterior alternating angles between parallel lines are congruent.

Prove:

Interior alternating angles  and exterior alternating angles are congruent (that is, they have the same measure of the angle.)

How to approach this problem?

So how do we go about this? We’ve already proven a theorem about 2 sets of angles that are congruent. So we will try to use that here, since here we also need to prove that two angles are congruent.

So let’s proceed to the proof, using what we already know about corresponding angles formed by a transverse line and two parallel lines, and the theorem we’ve already proven.

Alternate Interior Angles Theorem

Problem

AB||CD, prove m∠5 = m∠3 , and that m∠1 = m∠7 

Proof

Here's how you prove the alternate interior angles theorem:

(1)    AB||CD                    //given

(2)    ∠1 ≅ ∠5                   //from the axiom of parallel lines – corresponding angles

(3)    m∠1 = m∠5                           //definition of congruent angles 

(4)    m∠1 = m∠3                           //vertical, or opposite angles

(5)    m∠5 = m∠3                            //using (3) and (4) and  transitive property of equality, both equal m∠1

And  in a similar way,

(1)    AB||CD                     //given

(2)    ∠1 ≅ ∠5                    //from the axiom of parallel lines – corresponding angles

(3)    m∠1 = m∠5                           //definition of congruent angles 

(4)    m∠5 = m∠7                           // vertical, or opposite angles

(5)    m∠1 = m∠7                            //using (3) and (4) and  transitive property of equality, both equal m∠5

The converse of this theorem is also true, that is, if m∠5 = m∠3 , or if m∠1 = m∠7  , then AB||CD.

« Consecutive Interior Angles Theorem
Converse of the Corresponding 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]

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