Vertical Angles Theorem

The Theorem

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 –  (∎).

How to solve similar geometry problems

Ok, great, I’ve shown you how to prove this geometry theorum. 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.