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### Course: High school geometry>Unit 3

Lesson 4: Theorems concerning triangle properties

# Triangle exterior angle example

An exterior angle is formed outside a triangle by extending a side. Its measure is equal to the sum of the two non-adjacent interior angle measures, thanks to the fact that all angle measures in a triangle add up to 180 degrees. Created by Sal Khan.

## Want to join the conversation?

• Does anyone have a way to remember what supplementary and complementary mean? I keep mixing the two up when doing problems. (I mean, like a catchy phrase or something...)
• You get a COMPLEMENT when you do something RIGHT ! some complementary angles group to form a right angle (90 degreees).
• Okay uhh when he told me to pause the video i did and worked out the problem. it took me like 30 seconds but I ended up getting the same answer even though I did it a totally different way. I took two of the bottom angles 64 and 31 and I added them, which got 95. Then I subtracted that from 180 and got 85. I assumed that 85 was the whole top angle (the 50 degree angle and the unknown one). So I subtracted the known 50 degree angle from that and got 35 which was the answer. Don't know if I'm doing it wrong but do you have to do it the other way? It looks more confusing.
• the way you do it is just fine. there are just multiple ways.
• At why can't you just do this:
31+50+64+x=180
x=35
• Because connecting these two triangles would mean making a bigger triangle and the angle measurements would have to add up to 180°, this would be a way to solve this problem as well. Doing this would allow you to find the missing measurement of other part of the third angle.
• How do i remember the difference between supplementary and complementary
• Just remenber that you get compliments when you get things RIGHT! complementry angles add up to form a right angle
• I can't quite wrap my head around the idea that a triangle can have 3 angles with different degrees. Does anyone know how that is possible?
• The total of a triangles degrees is 180. With 3 intersecting sides this is possible in any number of combinations as long as the combinations of the 3 angles add up to 180 degrees.
• Do the angles of a triangle always add up to 180 degrees?
(this is prob a stupid question)
• They always add up to 180 degrees.

By the way, no question is stupid! It's just you requesting information!
• for the original triangle, couldn't you just do 180-50-64-31 and get 35?
• Yes, you can do that, but I suppose we wanted to demonstrate using triangle exterior angle here, so we don't do that
• Why is it that the Angles of a triangle add up to 180?
• An intuitive way to show this is to cut out a triangle from paper. Rip off the corners and put them together with the points of the triangle touching. If you make them all adjacent to one another, they will form 180 degrees.

To prove this, let's say we have a triangle ABC. We could create a line parallel to BC through point A. Using the angle pair relationships of parallel lines, we would find that there are two pairs of alternate interior angles that are congruent to each other. By doing this, we show that the two additional angles that were formed when we drew in the parallel line and the original interior angle A all form 180 degrees and that those two new angles are congruent to the two original interior angles at B and C.
• How do I re-watch a video?
• you either pruss the rewind button or if you're on a question it should be there if you scroll down, hope that helps! :)
(1 vote)
• At ,does it always right
• why odes a triangle absolutely have to have 180 degrees altogether (*__*)

## Video transcript

What I want to do now is just a series of problems that really make sure that we know what we're doing with parallel lines and triangles and all the rest. And what we have right here is a fairly classic problem. And what I want to do is I want to figure out, just given the information here-- so obviously I have a triangle here. I have another triangle over here. We were given some of the angles inside of these triangles. Given the information over here, I want to figure out what the measure of this angle is right over there. I need to figure out what that question mark is. And so you might want to give a go at it just knowing what you know about the sums of the measures of the angles inside of a triangle, and maybe a little bit of what you know about supplementary angles. So you might want to pause it and give it a try yourself because I'm about to give you the solution. So the first thing you might say-- and this is a general way to think about a lot of these problems where they give you some angles and you have to figure out some other angles based on the sum of angles and a triangle equaling 180, or this one doesn't have parallel lines on it. But you might see some with parallel lines and supplementary lines and complementary lines-- is to just fill in everything that you can figure out, and one way or another, you probably would be able to figure out what this question mark is. So the first thing that kind of pops out to me is we have one triangle right over here. We have this triangle on the left. And on this triangle on the left, we're given 2 of the angles. And if you have 2 of the angles in a triangle, you can always figure out the third angle because they're going to add up to 180 degrees. So if you call that x, we know that x plus 50 plus 64 is going to be equal to 180 degrees. Or we could say, x plus, what is this, 114. X plus 114 is equal to 180 degrees. We could subtract 114 from both sides of this equation, and we get x is equal to 180 minus 114. So 80 minus 14. 80 minus 10 would be 70, minus another 4 is 66. So x is 66 degrees. Now, if x is 66 degrees, I think you might find that there's another angle that's not too hard to figure out. So let me write it like this. Let me write x is equal to 66 degrees. Well if we know this angle right over here, if we know the measure of this angle is 66 degrees, we know that that angle is supplementary with this angle right over here. Their outer sides form a straight angle, and they are adjacent. So if we call this angle right over here, y, we know that y plus x is going to be equal to 180 degrees. And we know x is equal to 66 degrees. So this is 66. And so we can subtract 66 from both sides, and we get y is equal to-- these cancel out-- 180 minus 66 is 114. And that number might look a little familiar to you. Notice, this 114 was the exact same sum of these 2 angles over here. And that's actually a general idea, and I'll do it on the side here just to prove it to you. If I have, let's say that these 2 angles-- let's say that the measure of that angle is a, the measure of that angle is b, the measure of this angle we know is going to be 180 minus a minus b. That's this angle right over here. And then this angle, which is considered to be an exterior angle. So in this example, y is an exterior angle. In this example, that is our exterior angle. That is going to be supplementary to 180 minus a minus b. So this angle plus 180 minus a minus b is going to be equal to 180. So if you call this angle y, you would have y plus 180 minus a minus b is equal to 180. You could subtract 180 from both sides. You could add a plus b to both sides. So plus a plus b. Running out of space on the right hand side. And then you're left with-- these cancel out. On the left hand side, you're left with y. On the right hand side is equal to a plus b. So this is just a general property. You can just reason it through yourself just with the sum of the measures of the angles inside of a triangle add up to 180 degrees, and then you have a supplementary angles right over here. Or you could just say, look, if I have the exterior angles right over here, it's equal to the sum of the remote interior angles. That's just a little terminology you could see there. So y is equal to a plus b. 114 degrees, we've already shown to ourselves, is equal to 64 plus 50 degrees. But anyway, regardless of how we do it, if we just reason it out step by step or if we just knew this property from the get go, if we know that y is equal to 114 degrees-- and I like to reason it out every time just to make sure I'm not jumping to conclusions. So if y is 114 degrees, now we know this angle. We were given this angle in the beginning. Now we just have to figure out this third angle in this triangle. So if we call this z, if we call this question mark is equal to z, we know that z plus 114 plus 31 is equal to 180 degrees. The sums of the measures of the angle inside of a triangle add up to 180 degrees. That's the only property we're using in this step. So we get z plus, what is this, 145 is equal to 180. Did I do that right? We have a 15, then a 30. Yep, 145 is equal to 180. Subtract 145 from both sides of this equation, and we are left with z is equal to 80 minus 45 is equal to 35. So z is equal to 35 degrees, and we are done.