 Take a minute and sketch this example in your notes. Did you do it? Is it sketched in there? Good. Alright, so we want to find angle 1 and angle 2. Angle 1. Angle 1 is an exterior angle to triangle ABC. And as a result, I know that the exterior angle theorem says 1 is equal to the sum of the two remote interiors. Now the two remote interior angles are the ones that don't touch angle 1. In other words, the two remote interiors would be 35 and 25. So once again, the exterior angle theorem says angle 1 is equal to 25 plus 35, which makes 60 degrees. Lastly, we need to find angle 2. And angle 2, well, there are different ways to go about finding it. One possibility is to look at triangle ABC and triangle ABC using the triangle sum theorem. The three angles have to add up to 180. So 25 plus 35 plus angle 2 has to add up to 180. So that's one way to get the fact that the measure of angle 1 is 120. Another way, if you want, another way is to look at angles 1 and 2 together. And they form a linear pair. Since angle 1 is equal to 60, I know angle 2 must equal 120 because linear pairs are supplementary. In other words, they form 180 degrees. Here's another example dealing with exterior angle theorem. We've got the exterior angle, 145, and it's remote interior angles. Again, those are the ones that don't touch the angle. The remote interiors are 95 and 2x. So the exterior angle theorem says the sum of the two remote interior angles, so 2x and 95, together they add up to 145. So if we subtract 95 from both sides, we see 2x is equal to 50, which means x in this case would be 25. So that means that this angle would be 2x or 2x25, so it would be 50 degrees. This last example, it's a good one. Take a second, you're going to want to write this down. Once again, sketch this out in your notes. We've got a bunch of different angles to find. It's like a little puzzle. So let's go piece by piece. First off, let's take a look at angle 1. Angle 1 is an exterior angle to this triangle, this one on the far left. Its remote interior angles are 80 and 60, and so that means angle 1 is the sum of the remote interior angles, or 140 degrees. Angle 2 forms a linear pair with angle 1, and so that means angle 2 must be equal to 40 degrees, since 40 plus 180 adds up to 180 degrees. Now to find angle 3. To find angle 3, we're going to use the exterior angle theorem once again. So take a look at this triangle. In this triangle, the red one, I see 105 is an exterior angle, and the remote interiors are 40 degrees and angle 3. So that means the sum of the two remote interior angles is equal to the exterior. And so we can solve, measure angle 3 is whatever 105 minus 40 is. So angle 3 is 65 degrees. Now to find, let's say we want to find angle 4 next, I see 4 and 105 degrees form another linear pair. So that means angle 4 must be 75 degrees. So I've got 75 is angle 4, angle 40 degrees. So in this last triangle, angle 5 is another exterior angle, and the exterior angle is the sum of the two remote interiors. So 40 and 75, together angle 5 should be 115 degrees.