Class 9 math (India)
Find angle measures in triangles when the given measures are algebraic expressions. The key idea is that the sum of all angles in a triangle is always 180 degrees. We can also use parallel lines and transversals to find angle measures. Created by Sal Khan.
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- I dont understand the second example at all... can someone help?(23 votes)
- He does not tell you one crucial point: B, C, and D are collinear. So the line that contains B, C, and D is a transversal of the two parallel lines, etc.(7 votes)
- I'm having trouble with the star geometry problems. Could someone possibly break them down for me? I want to understand everything well, and that's the only thing giving me trouble.(5 votes)
- Oh, this is going to be hard to explain...
So, apologies if it isn't sufficient for you.
Here is what the problem looks like:
Create an upside-down pentagon.
The point of the pentagon (which is upside down) measures 105 degrees.
Create the star legs stemming from the pentagon. In the bottom left triangle, there is another given angle. It is the farthest angle down, and it measures 39 degrees.
Now to the left of the pentagon, is the angle we are trying to find, X.
(You may be able to find the exercise by just going through "Finding angle measures using Triangles" over and over. That actually may be easier than visualizing it...)
Anyway, to find X, we use some rules that are fundamental to geometry. A straight angle is equal to 180 degrees, specifically.
So, if it gives us 105, we know the angle(s) adjacent to it has to be 75. This is because a straight angle equals 180, adding the two must equal 180.
So we now have 75 degrees and 39 degrees in the bottom left triangle. This equals 114 in total. We know the interior angles of a triangle also equals 180 degrees. So if we already have the 114, that means the angle opposite of X, the vertical angle, is equal to 66. (Vertical angles are always equal to each other, so if one is 66, the other is 66.)
Thus we have solved for X.
I don't know if that helps, but I hope it does.
If you were stuck on another problem or one similar, just ask another question rather than continuing this strand of comments and answers. Anyway, good luck! Hope that helps!(9 votes)
- I still don't understand how to do it and I'm really stuck on these types of questions so can u go through some more examples with me?(8 votes)
- i didn't understand the 4x thing(5 votes)
- The 4x comes from translating words to Math. the measure of the largest angle (lets call L) is (=) 4 times (multiply) the second largest angle (which has been defined as x). So to translate this into Math, L= 4x.(5 votes)
- So in the first problem he did,why couldnt you substitute 4x for 4y and then do the problem from there?(3 votes)
- The goal is to set up an equation that will allow you to solve for a specific value. If you used 4y, you would get:
The only way to solve at this point would be to assume that y=x (which we could in fact do, based on the information in the question), then substitute one of the variables (i.e. y for x). Of course this would be the same as using only one variable to begin with:
- why are supplementary angles equal(5 votes)
- Supplementary angles are not equal. They sum up to 180 degrees. For example, 100 degrees and 80 degrees are supplementary.(4 votes)
- What does beta and theta mean?(4 votes)
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.