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

Lesson 6: Vertical, complementary, and supplementary angles- Identifying supplementary, complementary, and vertical angles
- Complementary & supplementary angles
- Complementary and supplementary angles (visual)
- Complementary and supplementary angles (no visual)
- Complementary and supplementary angles review
- Vertical angles
- Vertical angles review
- Angle relationships example
- Vertical angles are congruent proof

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# Vertical angles are congruent proof

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

## Want to join the conversation?

- what is orbitary angle.(97 votes)
- Did you mean an arbitrary angle? Because that is an angle that is undetermined, without a given measurement.(91 votes)

- can <DBA + <ABE= 180 DEGREE , Its important to write <DBA + <DBC = 180 DEGREE?

how did u prove dat <CBE = <DBA? NT GOT IT?(12 votes)- This is proven by the fact that they are "Supplementary" angles. By definition Supplementary angles add up to 180 degrees.

http://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/complementary-supplementary-angl/v/complementary-and-supplementary-angles

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.(15 votes)

- What makes an angle congruent to each other?(5 votes)
- 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.(19 votes)

- What is the purpose of doing proofs? Is it just the more sophisticated way of saying show your work?(7 votes)
- 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.(12 votes)

- 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?(4 votes)
- Usually, people would write a double curved line, but you might want to ask your teacher what he/she wants you to write.(7 votes)

- What is the difference between vertical angles and linear angles?(4 votes)
- 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.(5 votes)

- 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.(8 votes)

- What elements constitute a proof?(2 votes)
- A proof should include a
*given*(what you are "given" or what you are told), a*prove*statement (what you are trying to prove), and then a certain way of showing the steps you took to prove the prove statement from the given statement(s). Probably the most common of these ways is the two-column proof, which (as its name says) consists of two columns. The left column is the statements (what you derive from the given and from previous statements) and the right is the reasons (why you put the statements). The statements should be numbered and line up with their reasons. What Sal used was a**flow proof**, or a diagram showing the relationships between the statements to reach the prove statement.(4 votes)

- Why does the angles always have to match?(3 votes)
- Why didn't he just say that <CBE is vertical to <DBA, he said in the last video that all vertical angles are congruent.(2 votes)
- This video is to prove that all vertical angles are congruent hence the title.(2 votes)

## Video transcript

What I want to do in this video is prove to ourselves that vertical angles really are equal to each other, their measures are really equal to each other. So let's have a line here and let's say that I have another line over there, and let's call this point A, let's call this point B, point C, let's call this D, and let's call this right over there E. And so I'm just going to pick an arbitrary angle over here, let's say angle CB --what is this, this looks like an F-- angle CBE. What I want to do is if I can prove that angle CBE is always going to be equal to its vertical angle --so, angle DBA-- then I'd prove that vertical angles are always going to be equal, because this is just a generalilzable case right over here. So what I want to prove here is angle CBE is equal to, I could say the measure of angle CBE --you will see it in different ways-- actually this time let me write it without measure so that you get used to the different notations. I will just say prove angle CBE is equal to angle DBA. Is equal to angle DBA. So the first thing we know...the first thing we know so what do we know? We know that angle CBE, and we know that angle DBC are supplementary they are adjacent angles and their outer sides, both angles, form a straight angle over here. So we know that angle CBE and angle --so this is CBE-- and angle DBC are supplementary. I will just write "sup" for that. They are supplementary. Which means that angle CBE plus angle DBC is equal to 180 degrees. Fair enough. We also know --so let me see this is CBE, this is what we care about and we want to prove that this is equal to that-- we also know that angle DBA --we know that this is DBA right over here-- we also know that angle DBA and angle DBC are supplementary this angle and this angle are supplementary, their outer sides form a straight angle, they are adjacent so they are supplementary which tells us that angle DBA, this angle right over here, plus angle DBC, this angle over here, is going to be equal to 180 degrees. Now, from this top one, this top statement over here, we can subtract angle DBC from both sides and we get angle CBE is equal to 180 degrees minus angle DBC that's this information right over here, I just put the angle DBC on the right side or subtracted it from both sides of the equation and this right over here, if I do the exact same thing, subtract angle DBC from both sides of the equation, I get angle DBA is equal to 180 degrees --let me scroll over to the right a little bit-- is equal to 180 degrees minus angle DBC. So clearly, angle CBE is equal to 180 degrees minus angle DBC angle DBA is equal to 180 degrees minus angle DBC so they are equal to each other! They are both equal to the same thing so we get, which is what we wanted to get, angle CBE is equal to angle DBA. Angle CBE, which is this angle right over here, is equal to angle DBA and sometimes you might see that shown like this; so angle CBE, that's its measure, and you would say that this measure right over here is the exact same amount. And we have other vertical angles whatever this measure is, and sometimes you will see it with a double line like that, that you can say that THAT is going to be the same as whatever this angle right over here is. You will see it written like that sometimes, I like to use colors but not all books have the luxury of colors, or sometimes you will even see it written like this to show that they are the same angle; this angle and this angle --to show that these are different-- sometimes they will say that they are the same in this way. This angle is equal to this vertical angle, is equal to its vertical angle right over here and that this angle is equal to this angle that is opposite the intersection right over here. What we have proved is the general case because all I did here is I just did two general intersecting lines I picked a random angle, and then I proved that it is equal to the angle that is vertical to it.