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I thought I would do some more example problems involving triangles. And so in this first one it says the

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measure of the largest angle in a triangle is four times the measure of the second largest angle. The

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smallest angle is ten degrees. What are the measures of all the angles? Well we know one of them, we

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know it's ten degrees.

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Let's draw an arbitrary triangle right over here. now let's say that is our triangle.

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We know the the smallest angle is going to be ten degrees and I'll just say let's just assume that this

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right over here is the measure of the smallest angle. It's ten degrees. Now let's call the second largest

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angle, let's call that x. So this is going to be x. And then the first sentence, they say the measure

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of the largest angle in the triangle is four times, four times the measure of the second largest angle.

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So if the second largest angle is x, four times that measure is going to be 4x. So the largest angle

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is going to be 4x. And so, the one thing we know about the measures of the angles inside a triangle is

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they add up to 180 degrees. So we know that 4x + x + 10 degrees is going to be equal to 180 degrees.

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And 4x + x, that just gives us 5x. And then we have 5x + 10 is equal to 180 degrees. Subtract 10 from

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both sides, you get 5x = 170. And so, x = 170 / 5. Let's see, it will go into it 34 times? Let me verify

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this. So 5 goes into, yeah, it should be 34 times. It's going to go into it twice as many times as 10

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would go into it. 10 would go into 170 17 times, 5 would go into 170 34 times.

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So we can verify it. 3 times 5 is 15. Subtract, we get 2. Bring down the 0.

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5 goes into 20 four times. And then you can have the remainder. 4 times 5 is 20.

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No remainder. So it's 34 times. X is equal to 34.

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So the second largest angle has the measure of 34 degrees. This angle up here is going to be 4 times that.

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So 4 times 34, it's gonna be 120 degrees plus 16 degrees. This is going to be...

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136 degrees. Is that right? 4 times 4 is 16. 4 times 30 is 120.

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16 plus 120 is 136 degrees. So we're done.

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The three measures or the sizes of the three angles are 10 degrees, 34 degrees and 136 degrees.

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Let's do another one.

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So let's see. We have a little bit of a drawing here.

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And what I want to do is, and we could think about different things.

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We could say, let's solve for x. I'm assuming that 4x is the measure of this angle.

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2x is the measure of that angle right over there.

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We can solve for x and if we know x we can figure out what the actual measures of these angles are. Assuming that we can figure out x.

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And the other thing that they tell us is that this line over here is parallel to this line over here.

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And it's craftily drawn because it's parallel but one stops here and one sparks up there.

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So the first thing I want to do, if they're telling us these two lines are parallel,

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it's probably going to be something involving transversals or something.

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It might be something involving, the other option is something involving triangles.

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And at first you might say, wait, is this angle and that angle vertical angle?

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But we have to be very careful. They are not. These are not the same lines.

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This line is parallel to that line. This line it's bending right over there.

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So we can't make any attempt of assumption like that.

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So the interesting thing, and I'm not sure if this will lead us in the right direction,

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is to just make it clear that these two are part of parallel lines.

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So I could continue this line down like this. And I can continue this line up like that.

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And then that starts to look a little bit more like we're used to when we're dealing with parallel lines.

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And then this line segment BC or we could even say line BC if we were to continue it on,

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if we were to continue it on and on even pass D.

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Then this is clearly a transversal of those two parallel lines. This is clearly a trasnversal.

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And so if this angle right over here, if this angle right over here is 4x,

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it has a corresponding angle. Half of the..or maybe most of the work on all these

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is to try to see the parallel lines and see the transversal and see the things that might be useful for you.

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So that right there is the transversal.

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These are the parallel lines, that's one parallel line, that's the other parallel line.

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You can almost try to zone out all the other stuff in the diagram.

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And so if this angle right over here is 4x, it has a corresponding angle where the transversal intersects the other parallel line.

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This right here is its corresponding angle.

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So let me draw it in the same yellow.

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So this right here is the corresponding angle.

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So this will also be, this will also be 4x.

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And we see that this angle and this angle, this angle has a measure of 4x,

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this angle has a measure of 2x. We can see that they're supplementary.

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They're adjacent to each other. The outer sides form a straight angle.

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So they're supplementary which means that their measure add up to 180 degrees.

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They go all the way around like that, if you add the two adjacent angles together.

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So we know that 4x plus 2x needs to be equal to 180 degrees.

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Or we get 6x is equal to 180 degrees.

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Divide both sides by 6, you get x is equal to 30.

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And this angle right over here is 2 times x, so it's going to be 60 degrees.

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So this angle right over here is going to be 60 degrees.

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And this angle right over here is 4 times x. So it is 120 degrees.

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And we're done.