Question

How do you prove that vertical angles are congruent?

Answer 1

Vertical angles are angles that form when two lines cross, like the opposite sides of an "X". Suppose you have an "X", and the angle at the top is ...

Answer 2

Vertical angles are congruent: If two angles are vertical angles, then they’re congruent (see the above figure). http://media.wiley.com/Lux/27/220427.image0.jpg

Answer 3

When two lines cross to form an X, you easily see at least 4 angles. The up-down pair of angles (I'll call them v and ^) are 'vertical angles' and so are the right-left pair (I'll call them < and >). Someone could have come up with a better name than 'vertical' ... especially since 'vertical' makes you think of "standing up straight." Here, 'vertical' means "of the vertex." Since the lines cross and form a vertex, the angles get called vertical.

Now, why would the angles v and ^ be the same measure (congruent)? The easiest answer is "Because they both have the same supplement."

Which supplement do they have in common? You could say either < or >, it doesn't matter.

Of course, I'm being a bit lazy by not using geometry-standard terms, so now we must label everything correctly and formally:

Given two intersecting lines, AB and CD, whose intersection is at the point X. Prove that angle AXC is congruent to DXB. (Hopefully your drawing will match my setup!)

PROOF:

(1) Angle AXC is supplementary to angle CXB.
Reason: Angles AXC and CXB form a straight angle.

(2) Angle DXB is also supplementary to angle CXB.
Reason: (Same as Step 1)

(3) Angle AXC is congruent to angle DXB.
Reason: Two angles that are supplementary to the same third angle, are congruent.

Hope this sets you on your way!