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Home » Lines and Angles » Intersecting Lines and Angles » Vertical Angles Theorem

Vertical 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'll see a detailed step by step proof of the vertical angles theorem, which says that opposite angles of two intersecting lines are congruent. Plus, learn how to solve similar problems on your own!

Intersecting Lines in Geometry

The Theorem

The Vertical Angles Theorem states that the opposite (vertical) angles of two intersecting lines are congruent.

The problem

Prove: ∠1 ≅∠3 and ∠2 ≅ ∠4

Proof

(1)   m∠1 + m∠2 = 180°    // straight line measures 180° 

(2)   m∠3 + m∠2 = 180°    // straight line measures 180

(3)   m∠1 + m∠2 = m∠3 + m∠2          // transitive property of equality, as both left-hand sides of the equation sum up to the same value (180°  )

(4)   m∠1 = m∠3                           // subtraction property of equality (subtracted m∠2 from both sides)

(5)   ∠1 ≅ ∠3                            // definition of congruent angles

Similarly, for ∠2 ≅∠4:

(1)   m∠3 + m∠2 = 180°                      // straight line measures 180° 

(2)   m∠3 + m∠4 = 180°                      // straight line measures 180°

(3)   m∠3 + m∠2 = m∠3 + m∠4            // transitive property of equality, as both left  hand sides of the equation sum up to the same value (180°  )

(4)   m∠2 = m∠4                              // subtraction property of equality (subtracted m∠3 from both sides)

(5)   ∠2 ≅ ∠4                               // definition of congruent angle

And thus we have proven the theorem.

Quod erat demonstrandum

Often, you will see proofs end with the latin phrase "quod erat demonstrandum”, or QED for short, which means “what had to be demonstrated” or “what had to be shown”. There is also a special charter sometimes used -  (∎).

Strategy: How to solve similar problems

Ok, great, I’ve shown you how to prove this geometry theorem. But suppose you are now on your own –how would you know how to do this?

Well, in this case, it is quite simple. All we were given in the problem is a couple of intersecting lines. And the only definitions and proofs we have seen so far are that a line’s angle measure is 180°, and that two supplementary angles which make up a straight line sum up to 180°.

This problem has two sets of two supplementary angles which make up a straight line. So that’s the hint on how to proceed. Make use of the straight lines – both of them - and what we know about supplementary angles.

« Perpendicular Transversal Theorem
Consecutive Interior 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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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
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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