If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Geometry (all content)>Unit 2

Lesson 6: Vertical, complementary, and supplementary angles

# Vertical angles are congruent proof

Proving that vertical angles are equal. Created by Sal Khan.

## Want to join the conversation?

• what is orbitary angle.
• Did you mean an arbitrary angle? Because that is an angle that is undetermined, without a given measurement.
• can <DBA + <ABE= 180 DEGREE , Its important to write <DBA + <DBC = 180 DEGREE?
how did u prove dat <CBE = <DBA? NT GOT IT?
• This is proven by the fact that they are "Supplementary" angles. By definition Supplementary angles add up to 180 degrees.

To prove this imagine circle. Along time ago the ancient Babylonians defined a circle to be made up of 360 degrees. Now imagine we cut a line directly through the centre of the circle (a diameter). We have now cut our original 360 degrees in half giving us 180 degrees. So we know that the measure of an angle between any two points on a straight line is equal to 180 degrees.

Now imagine we drop another line perpendicular to our original line. This now cuts our 180 degrees in half and we now have two angles both measuring 90 degrees on either side of the the perpendicular line. Now imagine we rotate this line by say 10 degrees to the left. The angle on the right hand side of the line grows by ten degrees, and is now worth 100, and the angle on the left hand side shrinks by 10 degrees, and is now worth 80. notice that both angles still add up to 180 degrees.

This will always be true as no matter how much you rotate the line, in either direction, if you add to one angle you will always be subtracting that same amount from the other.

Hope that helped.
• What makes an angle congruent to each other?
• Congruent- identical in form; coinciding exactly when superimposed. This means they are they are put on top of each other, superimposed, that you could even see the bottom one they are 'identical' also meaning the same.
• What is the purpose of doing proofs? Is it just the more sophisticated way of saying show your work?
• Yes. There are informal and formal proofs. The ones you are referring to are formal proofs. They are steps all neatly organized to lead to a QED (proof) statement. Informal proofs are less organized. They are just written steps to more quickly lead to a QED statement. You could do an algebra problem with the T shape, like a formal proof, with the same idea. It is just to stay organized.
• Is it customary to write the double curved line or the line with the extra notch on the larger angle, or does that not matter?
• Usually, people would write a double curved line, but you might want to ask your teacher what he/she wants you to write.
• What is the difference between vertical angles and linear angles?
• Imagine two lines that intersect each other. That gives you four angles, let's call them A, B, C, D (where A is next to B and D, B is next to A and C and so on).

There are two pairs of vertical angles; A = C and B = D. They only connect at the very tip of the angles.

There are four linear pairs. Linear pairs share one leg and add up to 180 degrees. A&B, B&C, C&D, D&A are linear pairs.
• How do you know that <ABC is a line and not an angle that has a measure of say 179.999999?
(1 vote)
• If you used `∠BAC`, you already accepted it as an angle. Calm your imagination down.
• What elements constitute a proof?