Vertical angles are congruent is that an example of a theorem

Proof:

∠1and∠2∠1and∠2 form a linear pair, so by the Supplement Postulate, they are supplementary. That is,

m∠1+m∠2=180°m∠1+m∠2=180°.

∠2and∠3∠2and∠3 form a linear pair also, so

m∠2+m∠3=180°m∠2+m∠3=180°.

Subtracting m∠2m∠2 from both sides of both equations, we get

m∠1=180°−m∠2=m∠3m∠1=180°−m∠2=m∠3.

Therefore,

∠1≅∠3∠1≅∠3.

You can use a similar argument to prove that ∠2≅∠4∠2≅∠4.In the figure, 

∠1≅∠3∠1≅∠3 and ∠2≅∠4∠2≅∠4.