 In this video, I'm going to talk about applying the laws of deductive reasoning. So the laws of deductive reasoning are the law of detachment and the law of syllogism. So what I'm going to do is I'm going to follow the directions, draw a conclusion from the given information. So I'm going to read this out and see if there's any conclusions that I can come to based on this information. So given that if a polygon is a triangle, so if I have a shape and it is a triangle, so it's a three, it's got three sides, then it has three. Oh, OK. So if a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Makes sense. Not a quadrilateral. And then that says polygon P is a triangle. So what would I be able to conclude from that? If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. If this polygon P is a triangle, I can conclude that polygon P is not a quadrilateral. Quadrilateral. Polygon P is not going to be a quadrilateral. That's the conclusion I can come from that. Now, there's another way to see this. There is a little bit of a different way to see this. Notice here that we have some statements. If a polygon is a triangle, so we have some hypothesis. Here's a hypothesis P, so we'll start there. Then it has three sides. There's a Q there. If a polygon has three sides, there's Q again. Then it is not a quadrilateral. That's a new one, so we call that R. So notice I'm using P, Q, and then Q, R. This should ring a bell that this is the law of syllogism. This is the logical progression of the law of syllogism. So what we're saying here is that polygon P is a triangle. So actually, I'm going to write the law of syllogism up here. If P, then Q. If Q, then R, then the last statement is if P, then R, kind of going from the first to the last one. So notice here that polygon P is a triangle, which is actually P, which is the first hypothesis here. If polygon P is a triangle, then we can automatically jump to the very end. We can automatically assume that polygon P is not going to be a quadrilateral. All right, so that was kind of using the law of syllogism. That didn't directly say that we're using the law of syllogism. But I was able to figure that out by labeling all of my hypothesis and conclusions, my hypothesis and conclusions. All right, move it on. Next example, applying the laws of deductive reasoning. Again, drawing a conclusion from the given information. Given if the sum of the measures of two angles is 180 degrees, so I take two angles, add them up, and I get 180, then the angles are supplementary. OK, that makes sense. If two angles are supplementary, then they are not the angles of a triangle. Then this next statement says, if the measure of angle A is 135 degrees and the measure of angle B is 145 degrees, what conclusion can I draw from that? So again, just like the last one, I'm going to read through this one more time. If the sum of the measures of two angles is 180 degrees, then the angles are supplementary. If two angles are supplementary, they are not angles of a triangle. They are not angles of a triangle. OK, so looking at these two angles, 135 degrees and 45 degrees, if I add those two up, the sum of the measures of two angles, if I add those up, I do in fact get 180 degrees. So if these are 180 degrees, I can conclude that these two angles are not the angles of a triangle. Angle A and angle B are not angles, are not angles of a triangle. There we go. Yeah, I can come to that conclusion. Now again, I'm going to go back and label these so we can kind of see it a little bit different way. The sum of the measures of two angles is 180 degrees. That's my first statement. That's my hypothesis. Then the angles are supplementary. There's my second one. We'll call that Q. If two angles are supplementary, so we're using that supplementary word again. So there's another Q. They are not angles of a triangle. New one, so we call that R, not angles of a triangle. All right, angle A is 135 degrees and angle B is 45 degrees. Now it's not blatantly apparent. It doesn't just come out and say it, but it does give me these two angles, which are the measures, the sum of the measures of two angles is 180 degrees. That is this statement right here. So this is actually a P statement. It doesn't blatantly say it, but in fact, 135 degrees and 45 degrees are two angles that have a sum of 180 degrees. So once we follow that logical progression, we were starting with P, and then we can go to the very, very end. They are not the angles of a triangle. So we can say that these two angles, A and B, are not the angles of a triangle. All right, so that's the conclusions that we can draw from that. So again, this was a video on applying the laws of deductive reasoning. Those laws are the law of detachment and the law of syllogism. In these two examples, we just used the law of syllogism to make a conjecture to draw these conclusions.