Question 725785
A pair of angles whose sides form two lines is called vertical angles. 
Vertical angles are congruent and it is easy to prove. We just use the fact that a linear pair of angles are supplementary; that is their measures add up to 180°.


so, first draw lines {{{l}}} and {{{m}}} intersecting at point {{{P}}}

angles {{{1}}} (left of the point {{{P}}}) and {{{3}}}  (right of the point {{{P}}}) are vertical angles since their sides form lines {{{l}}} and {{{m}}}, similarly, angles {{{2}}} (above of the point {{{P}}})  and {{{4}}} (below of the point {{{P}}}) are vertical angles  for the same reason


Proof

We show that <  1 {{{congruent}}} <  3.

By definition, {{{m}}} <  {{{1}}} + {{{m}}} < {{{2}}} = {{{180}}}° because linear pair of angles are supplementary.

Then {{{m}}}< {{{2}}} + {{{m}}} < {{{3}}} = {{{180}}}° for same reason: linear pair of angles are supplementary.



using substitution property of equality, we have
 {{{m}}} <  {{{1}}} + {{{m}}} < {{{2}}} = {{{m}}} <  {{{2}}} + {{{m}}} < {{{3}}} ; that is {{{180}}}° = {{{180}}}°.

Subtracting {{{m}}} < {{{2}}} from both sides, we have

{{{m }}}< {{{1}}} ={{{ m }}}< {{{3}}}.

Therefore, vertical angles are {{{congruent}}}.