The Alternate Interior Angle Theorem (Trigonometry)

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, our goal is 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 on the plane, okay? Now, one condition that we can 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 that 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, it horizontal, vertical, it has nothing to do with anything like that. So, this is 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 Well, the vertical angles are congruent to each other. Why is that? Well, notice that you have angle ABE right here, and you have angle ABC. So, you look at these two angles, they're not necessarily congruent to each other. It looks that looks like ABC is bigger than ABE, 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°. Because they are supplementary. But, from a different perspective, if you take angle DBC and you take angle ABC, these angles together are also supplementary. These are also supplementary angles. In which case, if we then call this new angle DBC, 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°. 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 and the same thing. So, vertical angles have the same measure, therefore they are congruent. Another condition that guarantees uh congruent angles is when you have parallel lines, which are intersected by a transversal. So, what it 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 we know from perhaps past geometric experience that two lines in the plane are parallel if and only if they have equal slopes. Absolutely. Uh 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 antonym of parallel, uh they intersect at a right angle. The The picture that indicates perpendicular perpendicular lines looks like a pair of perpendicular lines. We do the same thing with parallel lines. Super clever uh 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 the 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, uh 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 uh 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 uh 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 uh the strip of the plane that's formed by the parallel band, uh you'll notice that things outside outside of the strip will be calling those exterior angles. Those Those inside of the band will 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. Um 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. Um 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 uh 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. Uh consecutive here means they're right next to each other uh like so. They're on the same side of the line. Uh 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 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. Uh and then finally, angles 1 and angle 5 are what we call corresponding angles. Okay, 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, one and eight, excuse me, one and five are corresponding, two and six are corresponding, three and seven are corresponding, and then four and eight 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 two right here, one is its supplement, three is also its supplement, and four is a vertical angle to two. If you look at six, six is the corresponding angle to two. Uh you get that five is the consecutive exterior angle to two. Uh you get that eight is the alternate exterior angle, and you don't actually have a name for two and seven right here. Uh so, I guess there is one that's missing. So, with regard to this diagram of the eight angles we talked about before, there's an important result in geometry that relates these things together. This is called the alternate interior angle theorem. Which the alternate interior angle theorem tells us if you 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 one and seven, which are alternate interior angles, this says that one and seven are congruent to each other. So, angle one is congruent to angle seven. But likewise, six and four are alternate interior angles, so they're going to be congruent as well. So, we get that angle four is congruent to angle six. But notice that angle one and angle three are vertical angles, so one and three are congruent to each other as well. So, you get that angle one is congruent to three, but angle seven will also be congruent to angle three. And likewise, five and seven are are vertical angles, so seven and five are congruent. And continuing on with this, we see that four and two are vertical angles, so they're congruent. Eight and six are congruent angles as well. And so then you end up with four is congruent to six, which is congruent to two, which is congruent to eight. Like so. And so with these with this transversal diagram, you get two families of congruence. You get the odd ones, one, three, five, and seven. They're all congruent to each other. And you get the even ones, two, four, six, eight. Who do we appreciate? Those are all congruent to each other as well. So with this with this pair of parallel lines with a transversal, we do get these congruent pairs. Um the ones that aren't congruent to each other are going to be supplementary. You'll notice that angles one and two are supplementary angles. Two is supplementary to three. And so by transitivity of congruence, we get that two is also supplementary to seven and five. So when you have these diagrams by the alternate interior angle theorem, either the angles are congruent or they're supplements. Those are the only options you get. I should also mention that this converse of the alternate interior angle theorem applies in geometry. That is to say, if you have a pair of alternate interior angles that are congruent, then the lines must be parallel. Uh that is to say, let me clean this up a little bit. That is to say, if the alternate interior angles one and seven happen to be congruent, then that would imply the angles were parallel. The congruence of alternate interior 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 one, two, three, and four given this diagram, which we know that the lines L and M are going to be parallel lines, and we know that angle We know that angle one is given as 3x + 2°. We also know that the measure of angle four is equal to 5x - 40 degrees like so. So what we can say here, what do we what do we know about these things? So, we know that angles one and two are supplementary angles. So, that means that the measure of angle one plus the measure of angle two is equal to 180°. We also know that two and three are supplements, but in particular one and three as vertical angles they're going to be equal to each other. So, that means that the measure of angle three is the same thing as well. What can we say about angle four? Okay? So, angle four is a corresponding angle to angle angle three. Are corresponding angles congruent to each other? Well, by the alternate interior angle theorem, yes. Because there's another angle over here, like angle five, angle five is an alternate interior angle to three, so they're congruent. And then five and four are congruent as well. So, the alternate interior angle theorem tells us that corresponding angles are congruent as well. So, angles one, three, and four are all congruent to each other. In particular, angle four is equal to well, this thing we saw right here. So, this gives us an equation we can work with. We get that 3x + 2 is equal to 5x - 4. And so, using this 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 two, we get that x equals 21. And so, now we can put all these things together. So, we see that the measure of angle one is equal to, like we said, 3 * 21 + 2 degrees. For which you get 3 * 21, that's going to be 63 + 2. So, we see that angle one is 65°. Now, 65° is also the measure of angle three. It's also the measure of angle four. And so the only one left to discover is angle two, but angle two is the supplement of angle one. So we see that the measure of angle two is going to equal 180° take away the measure of angle one, which was 65°. Therefore, angle two will measure to be 115°. So using congruent statements like alternate interior angles are congruent, corresponding 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.