- Angles in a triangle sum to 180° proof
- Find angles in triangles
- Isosceles & equilateral triangles problems
- Find angles in isosceles triangles
- Triangle exterior angle example
- Worked example: Triangle angles (intersecting lines)
- Worked example: Triangle angles (diagram)
- Finding angle measures using triangles
- Triangle angle challenge problem
- Triangle angle challenge problem 2
- Triangle angles review
Example of angle hunting! Created by Sal Khan.
Thought I would do some more example problems involving triangles. And so this first one, it says the measure of the largest angle in a triangle is 4 times the measure of the second largest angle. The smallest angle is 10 degrees. What are the measures of all the angles? Well, we know one of them. We know it's 10 degrees. Let's draw an arbitrary triangle right over here. So let's say that is our triangle. We know that the smallest angle is going to be 10 degrees. And I'll just say, let's just assume that this right over here is the measure of the smallest angle. It's 10 degrees. Now let's call the second largest angle-- let's call that x. So the second largest angle, let's call that x. So this is going to be x. And then the first sentence, they say the measure of the largest angle in a triangle is 4 times the measure of the second largest angle. So the second largest angle is x. 4 times that measure is going to be 4x. So the largest angle is going to be 4x. And so the one thing we know about the measures of the angles inside of a triangle is that they add up to 180 degrees. So we know that 4x plus x plus 10 degrees is going to be equal to 180 degrees. It's going to be equal to 180. And 4x plus x, that just gives us 5x. And then we have 5x plus 10 is equal to 180 degrees. Subtract 10 from both sides. You get 5x is equal to 170. And so x is equal to 170/5. And let's see, it'll go into it-- what is that, 34 times? Let me verify this. So 5 goes into-- yeah, it should be 34 times because it's going to go into it twice as many times as 10 would go into it. 10 would go into 170 17 times. 5 would go into 170 34 times. So we could verify it. Go into 170. 5 goes into 17 three times. 3 times 5 is 15. Subtract, you get 2. Bring down the 0. 5 goes into 20 four times, and then you're not going to have a remainder. 4 times 5 is 20. No remainder. So it's 34 times. So x is equal to 34. So the second largest angle has a measure of 34 degrees. This angle up here is going to be 4 times that. So 4 times 34-- let's see, that's going to be 120 degrees plus 16 degrees. This is going to be 136 degrees. Is that right? 4 times 4 is 16, 4 times 3 is 120, 16 plus 120 is 136 degrees. So we're done. The three measures, or the sizes of the three angles, are 10 degrees, 34 degrees, and 136 degrees. Let's do another one. So let's see. We have a little bit of a drawing here. And what I want to do is-- and we could think about different things. We could say, let's solve for x. I'm assuming that 4x is the measure of this angle. 2x is the measure of that angle right over there. We can solve for x. And then if we know x, we can figure out what the actual measures of these angles are, assuming that we can figure out x. And the other thing that they tell us is that this line over here is parallel to this line over here. And it was very craftily drawn. Because it's parallel, but one stops here, and then one starts up there. So the first thing I want to do-- if they're telling us that these two lines are parallel, there's probably going to be something involving transversals or something. It might be something involving-- the other option is something involving triangles. And at first, you might say, wait, is this angle and that angle vertical angles? But you have to be very careful. They are not. This is not the same line. This line is parallel to that line. This line, it's bending right over there, so we can't make any type of assumption like that. So the interesting thing-- and I'm not sure if this will lead in the right direction-- is to just make it clear that these two are part of parallel lines. So I could continue this line down like this. And then I can continue this line up like that. And then that starts to look a little bit more like we're used to when we're dealing with parallel lines. And then this line segment, BC-- or we could even say line BC, if we were to continue it on. If we were to continue it on and on, even pass D, then this is clearly a transversal of those two parallel lines. This is clearly a transversal. And so if this angle right over here is 4x, it has a corresponding angle. Half of the-- or maybe most of the work on all of these is to try to see the parallel lines and see the transversal and see the things that might be useful for you. So that right there is the transversal. These are the parallel lines. That's one parallel line. That is the other parallel line. You can almost try to zone out all of the other stuff in the diagram. And so if this angle right over here is 4x, it has a corresponding angle where the transversal intersects the other parallel line. This right here is its corresponding angle. So let me draw it in that same yellow. This right over here is a corresponding angle. So this will also be 4x. And we see that this angle-- this angle and this angle, this angle that has measure 4x and this angle that measures 2x-- we see that they're supplementary. They're adjacent to each other. Their outer sides form a straight angle. So they're supplementary, which means that their measures add up to 180 degrees. They kind of form-- they go all the way around like that if you add the two adjacent angles together. So we know that 4x plus 2x needs to be equal to 180 degrees, or we get 6x is equal to 180 degrees. Divide both sides by 6. You get x is equal to 30, or x is equal to-- well, I shouldn't say-- well, x could be 30. And then this angle right over here is 2 times x. So it's going to be 60 degrees. So this angle right over here is going to be 60 degrees. And this angle right over here is 4 times x. So it is 120 degrees, and we're done.