Finding more angles

Example of angle hunting!

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

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

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

know it's ten degrees.

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

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

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

So if the second largest angle is x, four 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 a triangle is

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

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

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

this. So 5 goes into, yeah, it should be 34 times. 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 can verify it. 3 times 5 is 15. Subtract, we get 2. Bring down the 0.

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

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

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

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

136 degrees. Is that right? 4 times 4 is 16. 4 times 30 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 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's craftily drawn because it's parallel but one stops here and one sparks up there.

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

it'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 angle?

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

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

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

So the interesting thing, and I'm not sure if this will lead us 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 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 trasnversal.

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

it has a corresponding angle. Half of the..or maybe most of the work on all 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's the other parallel line.

You can almost try to zone out all 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 the same yellow.

So this right here is the corresponding angle.

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

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

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

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

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

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.

And 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.