Hey guys, Mr. Backberg here. This is Lesson 1.2. In this video, we are going to use segment postulates to identify congruent segments. Now, lines go on forever, and because of that, we're not able to measure how long lines are, because it's impossible to measure something that just keeps going and going and going. But segments have endpoints.
Remember, segments stop at those endpoints, and because of that, we are able to measure how long a segment is. So what we're able to do is take the endpoints of a segment and line them up on a number line, or you're probably used to using a ruler to measure things. And then what we can do is, in order to find the distance between two points, what we do is we take the absolute value of the difference of the coordinates of each endpoint.
This is something called the ruler postulate. So if we're looking specifically at this line segment that's drawn out on our number line, and we wanted to figure out how long segment AC is, then based on this ruler postulate, what we're going to do is the absolute value, remember we show absolute value with two vertical bars, of the difference, difference means we're going to be doing some subtraction with the end coordinates. So we're looking at point A, which is negative 3, and point C, which is 6. Now if we take negative three minus six, we get negative nine, but there's still these absolute value bars around the outside, and remember, absolute value bars just make whatever number's inside of them positive.
So instead of having a negative nine, the answer would be positive nine. Now it doesn't matter what order we do this subtraction in. If we went 6 minus negative 3 instead, we'd end up with the absolute value of 9, because subtracting a negative is just like adding, and then the absolute value of 9 is just 9. So no matter which way we do it, A minus C or C minus A, we still end up with the same answer.
Now on this example, because it's on a number line, it might be really easy for us to just count spaces. So if we started with A and went... 1, 2, 3, 4, 5, 6, 7, 8, 9 spaces. That works out really nice, but sometimes when we're using a ruler or maybe our number line is set up a little bit differently, there might be fractions or there might be decimals, so this subtraction difference way can make things a little bit easier sometimes. One quick definition we'll need moving forward is the idea of points being between other points.
So if we're dealing with three collinear points, and that collinear piece is going to be really, really important to us. If we're dealing with three collinear points, then we can say that one point is between the other two. So if we look at this example on the left, we can say that point B is between points A and C, since those three points are collinear.
Now if we compare that to the one on the right, Point E is not considered to be between points D and F because point E is not on the same line as D and F. Now if we focus on that example where point B was between points A and C, then what happens to our segment is we've originally got this big long AC segment, but putting that point B between A and C splits that big segment into two smaller segments. we've got the segment that runs from A to B, and then we've got the segment that runs from B to C.
Well, there's this thing called the segment addition postulate, which says if we add up the length of the two smaller segments, so the length of AB and the length of BC, then that should equal the entire length from A to C. So we've got a couple examples that we're going to run through using the segment addition postulate. I'm going to write it out just so we don't forget what it is.
It says the length of AB plus the length of BC equals the length of AC. And we're given a picture to take a look at, just in case we forgot. So in this first example, it says the length of AB is 15, and the length of BC is 21. And what we want to do is we want to go through and find the length of AC. So I'm gonna use this segment addition postulate equation that we wrote down earlier to help us out. And what I'm gonna do is take the lengths that we have and just fill them in in the appropriate spot.
So it says the length of AB is 15. So first thing I'm gonna do is write down the 15 plus, now we know that the length of BC is 21, so we'll write that down, and we're trying to find AC. We don't know what that is, so we'll just leave the AC on the right-hand side alone. Now on the left-hand side, we've just got some addition going on. If we take 15 plus 21, we get 36. So 36 is the length from A to C. Now if we look at our next example, this time we're told that the length of AC is 42, the length of AB is 17, and we're going to figure out how long it is from B to C.
Just like we did above, I'm going to fill in the information in the appropriate spot. So it says that AC is 42. So on the right-hand side where we have our AC, I'm going to fill in 42. On the left-hand side, I'm going to fill in 17 for AB, plus we don't know what BC is, that's what we're trying to find. So I'm just going to leave BC alone. Now if we think about it algebraically, if we're trying to solve this equation, in order to move this 17 over to the other side, we would have to subtract 17 from both sides. The 17s on the left-hand side will cancel out, and all we'll have left over is the BC.
And then on the right-hand side, if we take 42 minus 17, we get 25. So here are two more examples. You can pause the video and run through them and then start it back up once you're all finished. So in this first one, if we're looking at using our segment addition postulate, it would go AB plus BC equals AC.
So if we fill in our AB length, that's 12, plus our BC length is 19. That equals our AC length. If we take 12 plus 19, we get 31 as our AC length. Looking at the second example, again setting up our segment addition postulate, our AB length is 8, our BC length is X, and our AC length is 23. Now if we subtract the 8 over to the right-hand side, we get an X value of 15. In this example, it says that point M is between points L and N. So what I want you to do is pause the video, draw a sketch of what this would look like, and then write out what the segment addition postulate would look like using these points.
Once you're done, you can start the video back up. So if we draw this out, here's our line, and on the left-hand side, I'm going to put point L. On the right-hand side, I'm going to put point N, and then somewhere between those points is point M, since it's between L and N. Now, writing out our segment addition postulate, the first small piece is LM, plus the other small piece is MN, and if we add those together, the big long piece is LN.
Now here's what we've got. We know that the LM segment is 7y plus 9. We also know that the MN segment is 3y plus 4, and the big LN segment is 143. So what we're going to do is plug these values into our segment addition postulate, and then do a little bit of algebra solving for y. So first on the left hand side we've got lm and that's our 7y plus 9 plus now we've got mn which is 3y plus 4 and that equals our ln piece which is 143. Now on the left hand side we're going to do some combining like terms. We've got 7y and 3y. If we add those together we've got 10y and if we take 9 plus 4 we get 13 and then the right hand side we've still got 143. Now if we subtract the 13 over to the right hand side we've got 10y equals 130 and then last step in order to get rid of the 10 we are going to divide both sides.
divides by 10, so we get a y value of 13. Now one idea that's really important in geometry is the idea of things being congruent. And the word congruent means the same size. So if we're talking about congruent segments, then these are multiple segments that have the exact same length.
Now notice there's a little bit of a difference between the way these things are written. The ones on the left-hand side don't have the bar over the top of them. So what this means is we're talking about the length of these segments.
So this would be the length of segment AB, and this would be the length of of segment AD. If we're talking about their lengths or their measurements, then we would say that those things are equal. But if we're talking about the segments themselves, so segment AB and segment AD, we use this symbol in the middle, which is an equal sign with a little squiggly line over the top of it, that's our congruent symbol. So if we're talking about measurements, those things are equal, but if we're talking about the segments themselves, then we would say that those things are congruent.
So in this example, we've got a couple of segments drawn out on a coordinate grid, and we want to find the length of each segment, so the length of JK and the length of LM, and then we're going to determine if those two segments are congruent. So let's look at the segment JK first. Now we should notice that this is a flat horizontal line.
In order to figure out its length, since it's a horizontal line, we're going to focus on the x value from each ordered pair. because x values move us left and right. So finding the length of segment JK, remember what we're gonna do is the absolute value of the difference in those two coordinates. So if we take negative three minus two, we get negative five, and the absolute value of negative five is five.
So the length of JK is five. Now if we're looking at segment LM, this one is a vertical. up and down segment, so we're going to be focusing on the y values from the ordered pairs, because y values move us up and down. So finding the length of lm, again we're going to do the absolute value of the difference in those coordinates, 3 minus negative 2, well double negative makes this addition, so we've got the absolute value of 5, which is just 5. So the length of jk is 5, the length of lm is 5, so we can say yes.
That segment that runs from J to K is congruent to the segment that runs from L to M. That's going to be it for this video. Thanks for watching.