Question

How do you prove vertical angles are congruent?

Answer 1

I'd like to try making this one simple, citing as few theorems as possible. First, I will just say it "in English" for you.

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.

Answer 2

OK! Thanks so much

Answer 3

No problem, have a great day!

Answer 4

@Veelution PLease help with my Question! :)