 Welcome back to our lecture series Math 1060, Trigonometry for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture three, we want to talk about relationships that exist between angles, and in particular goals to describe under what situations can we expect a triangle to be the same, two triangles to be the same, or under what situations can we expect two triangles to be similar. In this first video, we're going to talk about what conditions can guarantee that angles are the same, or more specifically that is the proper term in geometry is when are two angles congruent. We say that two angles are congruent if they have the same angle measure, and congruent angles will be denoted in diagrams. In a manner similar to this, we'll draw little arches, little arcs on them, the same number of arcs, the same color will indicate that those angles are congruent, they have the same angle measure. So for all intents and purposes, congruent angles are the same angle, even though they might exist in different places on the plane. Now, one condition that we have between angles that you can see in the diagram that guarantees angles are congruent is the idea of vertical angles. What does it mean for two angles to be vertical? Well, let's imagine we have two straight lines to intersect each other at some common point B. So B is the intersection of these two lines. Then we say that two angles are vertical if they're opposite of each other with respect to this intersection of the lines. So we have one set of vertical lines right here, and we have another set of vertical lines right here. So given intersection of two, two lines, you always get these two pairs of vertical lines. Not sure why they're exactly called vertical, right? I mean, horizontal vertical that has nothing to do with anything like that. It's just the terminology. The reason why vertical angles are important in this conversation of congruence is that vertical angles are necessarily congruent to each other. And let me kind of explain why that is. So I claim that these two angles here in red are vertical, the vertical angles are congruent to each other. Why is that? Well, notice that you have angle A, B, E right here, and you have angle A, B, C. So you look at these two angles, they're not necessarily congruent to each other. It looks that looks like A, B, C is bigger than A, B, E, but these two angles put together are in fact supplementary angles. All right? They're supplementary. So what that tells us is if we call the first, well, you know, if we call this one angle one and this angle two, then we get that the measure of angle one plus the measure of angle two adds up to be 180 degrees because they are supplementary. But from a different perspective, if you take angle D, B, C, and you take angle A, B, C, these angles together are also supplementary. These are also supplementary angles. In which case, if we then call this new angle D, B, C, we'll call it angle three. This tells us that the measure of angle three plus the measure of angle two is equal to 180 degrees. In particular, the measure of angle one plus the measure of angle two is equal to the measure of angle three plus the measure of angle two. And if you subtract the measure of angle two from both sides, you end up that angle one and angle two have the exact same measure. And so therefore we say that the angle one is congruent to angle two, excuse me, angle three. And so for congruent, you draw an equal sign with this little extra squiggle on top. Angle one and angle two are congruent to each other because their measures are one of the same thing. So vertical angles have the same measure, therefore they are congruent. Another condition that guarantees congruent angles is when you have parallel lines, which are intersected by a transversal. So remember, what does it mean for two lines to be parallel? Two lines in the plane are considered parallel if they're not intersecting. There's no point where the lines intersect each other. So it's like these two sides of a railroad just going on and on and on forever, never intersecting each other. And so we know from perhaps past geometric experience that two lines in the plane are parallel if and only if they have equal slopes, absolutely. If two lines say L and M are parallel, that's often denoted by this. The parallel symbol itself looks like two parallel lines, which is why we suggest that. Kind of like with perpendicular lines, which is like the antonoma parallel, the intersect the right angle. The picture that indicates perpendicular perpendicular lines looks like a pair of perpendicular lines. We do the same thing with parallel lines. Super clever, mnemonic device right there. So if two parallel lines are cut by a transversal, so a transversal will be a line that intersects both parallel lines. So you can see that in diagram. We have a line L right here, a line M, and then call this line T. T is a transversal to the parallel lines L and M. So with this type of diagram, you have two parallel lines and a transversal. There are eight angles formed by the intersection of the transversal with L and with M. So if we look at the angles formed by the transversal T intersecting the parallel line L, we'll label those angles 1, 2, 3, 4, as you see here. And then the four angles formed by intersecting T with M, we'll call those ones 5, 6, 7, 8, okay? So the reason we label these is so we can talk about some families, some pairs of angles that come from this diagram. So we say that angles 1 and 7 are alternate interior angles, interior in the respect that they're kind of inside the strip of the plane that's formed by the parallel band. You'll notice that things outside, outside of the strip, we'll be calling those exterior angles, those inside of the band we'll call interior angles, all right? So that's part of the name there. Alternate means they're on the opposite side of T. So 1 and 7 are on opposite sides of T. So 1 and 7 are considered alternate interior angles. Likewise, 4 and 6 are alternate interior angles. They're on the opposite side of the transversal, but they're interior to the parallel band. On the other hand, if we take for example the angles 2 and 8, these are called alternate exterior angles. Notice that they're alternate because they're on the opposite side of the transversal, and they're now exterior to this parallel band. They're not on the inside, they're on the outside of the picture. So 2 and 8 are alternate exterior angles. Similarly, 3 and 5 are alternate exterior angles. If we take angles 1 and 6, we call those ones consecutive interior angles. Interior makes sense because it's inside of the parallel band. Consecutive here means they're right next to each other, like so. They're on the same side of the line. Likewise, angles 4 and 7 are consecutive interior angles. By analog, angles 2 and 5 are consecutive exterior angles because they're consecutive meaning they're on the same side of the transversal. Exterior means they're outside of the parallel band. Likewise, 3 and 8 are consecutive exterior angles. Finally, angles 1 and angle 5 are what we call corresponding angles. Corresponding angles mean if you kind of think of the intersections of the transversal with one parallel line, and you look at the angles with the transversal with the other parallel line, if you look at the corresponding positions, so for example 1 and 8, excuse me, 1 and 5 are corresponding, 2 and 6 are corresponding, 3 and 7 are corresponding, and then 4 and 8 are corresponding. So with that, we can describe every pair of angles with one fun little name or another. So for example, if you take angle 2 right here, 1 is its supplement, 3 is also its supplement, and 4 is a vertical angle to 2. If you look at 6, 6 is the corresponding angle to 2. You get that 5 is the consecutive exterior angle to 2. You get that 8 is the alternate exterior angle, and you don't actually have a name for 2 and 7 right here, so I guess there is one that's missing, but I edited it, but I just write down things to say. So start again in 3, 2, 1. So with regard to this diagram of the 8 angles we talked about before, there's an important resultant geometry that relates these things together. This is called the alternate interior angle theorem, which the alternate interior angle theorem tells us if we have two parallel lines cut by a transversal, exactly like in this diagram, then alternate interior angles are congruent to each other. So for example, you take 1 and 7, which are alternate interior angles, this says that 1 and 7 are congruent to each other. So angle 1 is congruent to angle 7. But likewise, 6 and 4 are alternate interior angles, so they're going to be congruent as well. So we get that angle 4 is congruent to angle 6. But notice that angle 1 and angle 3 are vertical angles. So 1 and 3 are congruent to each other as well. So you get that angle 1 is congruent to 3, but angle 7 will also be congruent to angle 3. And likewise, 5 and 7 are vertical angles, so 7 and 5 are congruent. And continuing on with this, we see that 4 and 2 are vertical angles, so they're congruent. 8 and 6 are congruent angles as well. And so then you end up with 4 is congruent to 6, which is congruent to 2, which is congruent to 8, like so. And so with this transversal diagram, you get two families of congruence. You get the odd ones, 1, 3, 5, and 7, they're all congruent to each other. And you get the even ones, 2, 4, 6, 8, who do we appreciate? Those are all congruent to each other as well. So with this pair of parallel lines with the transversal, we do get these congruence pairs. The ones that aren't congruent to each other are going to be supplementary. You'll notice that angles 1 and 2 are supplementary angles, 2 is supplementary to 3. And so by transitivity of congruence, we get that 2 is also supplementary to 7 and 5. So when you have these diagrams by the alternative angle theorem, either the angles are congruent or they're supplements, if that's the only options you get. I should also mention that the converse of the alternative angle theorem applies in geometry. That is to say, if you have a pair of alternative angles that are congruent, then the lines must be parallel. That is to say, let me clean this up a little bit. That is to say, if the alternative angles 1 and 7 happened to be congruent, then that would imply the angles were parallel. The congruence of alternative angles is equivalent to the lines being parallel to each other. So let's put these principles to practice here. Let's find the measure of the following angles here. Let's find the measure of angles 1, 2, 3, and 4 given this diagram, which we know that the lines L and M are going to be parallel lines. And we know that angle 1 is given as 3x plus 2 degrees. We also know that the measure of angle 4 is equal to 5x minus 40 degrees like so. So what we can say here, what do we know about these things? So we know that angles 1 and 2 are supplementary angles. So that means that the measure of angle 1 plus the measure of angle 2 is equal to 180 degrees. We also know that 2 and 3 are supplements, but in particular 1 and 3 as vertical angles, they're going to be equal to each other. So that means that the measure of angle 3 is the same thing as well. What can we say about angle 4? So angle 4 is a corresponding angle to angle 3. Are corresponding angles congruent to each other? Well, by the alternative angle theorem, yes, because there's another angle over here like angle 5. Angle 5 is an alternate interior angle to 3, so they're congruent. And then 5 and 4 are congruent as well. So the alternate interior angle theorem tells us that corresponding angles are congruent as well. So angles 1, 3, and 4 are all congruent to each other. In particular, angle 4 is equal to, well, this thing we saw right here. So this gives us an equation we can work with. We get that 3x plus 2 is equal to 5x minus 4. And so using this connection, we can then solve for x. Let's subtract 3x from both sides. Let's add 40 to both sides. So we end up with 2x is equal to 42. So divide both sides by 2. We get that x equals 21. And so now we can put all these things together. So we see that the measure of angle 1 is equal to, like we said, 3 times 21 plus 2 degrees for which you get 3 times 21. That's going to be 63 plus 2. So we see that angle 1 is 65 degrees. Now 65 degrees is also the measure of angle 3. It's also the measure of angle 4. And so the only one left to discover is angle 2. But angle 2 is the supplement of angle 1. So we see that the measure of angle 2 is going to equal 180 degrees. Take away the measure of angle 1, which was 65 degrees. Therefore, angle 2 will measure to be 115 degrees. So using congruent statements like alternate interior angles are congruent, corresponded angles are congruent, and other consequences of the alternate interior angle theorem, we're able to solve for the missing angle measures on these angle diagrams. So it's very important we remember things like alternate interior angles and vertical angles when we consider various diagrams involving triangles and the like.