WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson is on proving lines parallel.
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We are actually going to take the theorems that we learned from the past few lessons, and we are going to use them to prove that two lines are parallel.
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We learned, from the Corresponding Angles Postulate, that if the lines are parallel, then the corresponding angles are congruent.
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If the lines are the parallel lines that are cut by a transversal, then the corresponding angles are congruent.
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Now, this one is saying, "If the two lines in a plane are cut by a transversal, and corresponding angles are congruent, then the lines are parallel."
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So, it is using the converse of that postulate that we learned a couple of lessons ago.
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And we are going to take that and use it to prove that lines are parallel.
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Before, when we used that postulate, it was given that the lines were parallel; and then you would have to show
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that the conclusion, "then the corresponding angles are congruent," would be true.
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But for this one, they are giving you that the corresponding angles are congruent.
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And then, the conclusion, "the lines are parallel," is what you are going to be proving.
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This first postulate: if you look at angles 1 and 2, those are corresponding angles.
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So, if I tell you that angle 1 and angle 2 are congruent, and they are corresponding angles, then the lines are parallel.
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As long as these two angles are congruent, then these lines are parallel; so I can conclude that these lines are parallel.
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Now, again, for them to be corresponding angles, the lines don't have to be parallel.
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Even if the lines are not parallel, they are still considered corresponding angles.
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But now, we know that, as long as the corresponding angles are congruent, then the lines will be parallel.
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The next postulate: If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.
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So again, this is the alternate exterior angles theorem's converse.
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The alternate exterior angles theorem said that, if two lines are parallel, then alternate exterior angles are congruent.
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This is the converse; so they are giving you that alternate exterior angles are congruent; then, the lines are parallel.
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Depending on what you are trying to prove, you are going to be using the different postulates--either the original theorem or postulate, or the converse, these.
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If you are trying to prove that the lines are parallel, then you are going to be using these.
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If you are trying to prove that the angles are congruent, then you are going to be using the original.
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So, alternate exterior angles are congruent; therefore, we can conclude that the lines are parallel.
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The Parallel Postulate: this is called the Parallel Postulate: If there is a line and a point not on the line,
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then there exists exactly one line through that point that is parallel to the given line.
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What that is saying is that I could only draw one line, a single line, through this point, so that it is going to be parallel to this line.
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I cannot draw two different lines and have them both be parallel to this line--only one.
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So, it would look something like that...well, that is not really through the point, but...
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this is the only line that I can draw to make it parallel to this line.
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Only one line exists; and that is the Parallel Postulate.
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Now, we are going to go over a few more theorems now.
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Before it was postulates, but now these are some theorems that we can use to prove lines parallel.
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If two lines in a plane are cut by a transversal (this is my transversal) so that a pair of consecutive interior angles is supplementary, then the lines are parallel.
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Remember consecutive interior angles? They are not congruent; they are supplementary.
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If you look at these angles, angle 1 is an obtuse angle; angle 2 is an acute angle.
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They don't even look congruent; they don't look the same.
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Make sure that consecutive interior angles are supplementary.
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Again, the original theorem had said, "Well, if the lines are parallel" (that is given), "then we can conclude that consecutive interior angles are supplementary."
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This one is the converse, saying that the given is that the consecutive interior angles are supplementary.
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Then, the conclusion is...we can conclude that these lines are parallel.
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If two lines in a plane are cut by the transversal, so that a pair of alternate interior angles is congruent, then the lines are parallel.
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So again, if these alternate interior angles are congruent, then the lines are parallel.
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And make sure that it is not the transversal; it is the two lines that the transversal is cutting through that are parallel.
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And the next theorem, the last theorem that we are going to go over today for this lesson:
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in a plane, if two lines are perpendicular to the same line, then they are parallel.
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See how this line is perpendicular to this line? Well, this is my transversal.
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So, if this line is perpendicular to this line, and this line is also perpendicular to that same line, the same transversal, then these lines will be parallel.
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If both lines are perpendicular to the same line, the transversal, then the two lines will be parallel.
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OK, let's go over a few examples: Determine which lines are parallel for each.
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This one right here, the first one, is giving us that angle ABC is congruent to angle (where is D?...) DGF.
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That means that this angle and this angle are congruent.
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OK, now again, when we look at angle relationships formed by the transversal, we only need three lines.
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We have a bunch of lines here; so I want to just try to figure out what three lines I am going to be using for this problem,
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and ignore the other lines, because they are just there to confuse you.
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This angle right here is formed from this line and this line, so I know that those two lines, I need.
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And then, this angle is also formed from this line and this line.
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So, it will be line CJ, line FN, and line AO; those are my three lines.
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This line right here--ignore it; this line right here--ignore it.
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We are only dealing with this line, this line, and this line.
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And from those three lines, we know that this line, AO, is the transversal, because that is the one that is intersecting the other two lines.
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So, if this angle and this angle are congruent, what are those angles--what is the angle relationship?
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They are corresponding; and the postulate that we just went over said that, if corresponding angles are congruent, then the lines are parallel.
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I can say that line CJ is parallel to line FN.
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The next one: angle FGO, this angle right here, is congruent to angle NLK.
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So again, I am using this line, this line, and this line, because it is this angle right here and this angle right here.
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So, for those two angles, their relationship is alternate exterior angles.
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If alternate exterior angles are congruent, then the two lines are parallel.
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And those two lines are going to be, since this FN is a transversal, line AO, parallel to HM.
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And remember that this is the symbol for "parallel."
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The measure of angle DBI (where is DBI?), this angle right here, plus the measure of angle BIK
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(BIK is right here) equals 180, so that these two angles are supplementary.
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Now, these two angles are consecutive interior angles, or same-side interior angles,
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which means that if they are supplementary (which they are, because 180 is supplementary), then the two lines are parallel.
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That is what the theorem says; so I know that line AO, just from this information, is parallel to HM.
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The theorem says that, if the two angles are supplementary, then the two lines are parallel.
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OK, the next one: CJ is perpendicular to HK; this is perpendicular, this last one.
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And then, FN is perpendicular to KN; there is the perpendicular sign.
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Remember the theorem that said that, if two lines are perpendicular to the same transversal, then the two lines are parallel.
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So then, from this information, I can say that CJ is parallel to FN.
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The next example: Find the value of x so that lines *l* and *n* will be parallel.
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I want to make it so that my x-value will make them have some kind of relationship, so that I can use the theorem,
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saying that I have to make two angles congruent or supplementary--something so that I can conclude that the lines are parallel.
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Let's see: these two angles right here don't have a relationship; this one is an interior angle, and this one is an exterior angle.
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But what I can do is use other angle relationships; if I use other angle relationships, then I can find some kind of relationship
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from the theorems or the postulates that we just went over.
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What I can do: since this angle and this angle right here are vertical angles, I know that vertical angles are congruent.
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And since vertical angles are always congruent, and these are vertical angles, since this is 4x + 13,
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I can say that this is also 4x + 13, because it is vertical, and vertical angles are congruent.
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Now, this angle and this angle have a special relationship, and that is that they are same-side or consecutive interior angles.
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I know that, if consecutive interior angles are (not congruent) supplementary, then the lines are parallel.
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As long as I can prove that these two are supplementary angles, then I can say that the lines are parallel.
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I am going to make 4x + 13, plus 6x + 7, equal to 180, because again, they are supplementary.
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Then, I am going to solve for x; this is going to be 4x + 6x is 10x; 13 + 7 is 20; 10x = 160, so x is going to be 16.
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So, as long as x is 16, then that is going to make these angles supplementary, and then the lines will be parallel.
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So, x has to be 16 in order for these two lines to be parallel.
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Find the values of x and y so that opposite sides are parallel.
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OK, that means that I want AB to be parallel to DC, and I want AD to be parallel to BC.
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So, the first thing is that...now, this is a little bit hard to see, if you want to think of it as parallel lines and transversals.
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So, what you can do: if you get a problem like this, you can make these lines a little bit longer.
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Extend them out, so that they will be easier to see.
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That means that this one right here and this one right here--if this is the line,
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and these are the two lines that the transversal is cutting through, then these two angles are going to be consecutive interior angles.
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This angle and this angle are consecutive interior angles.
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This angle and this angle are also consecutive interior angles, because it goes line, line, transversal.
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That means that these are same-side interior angles, or consecutive interior angles.
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Then, I have options: since I have the option of making this one and this one supplementary, I can also say that this one and this one are supplementary.
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But this one has the y, and this one has an x; so instead of using x and y to make it supplementary
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(you are going to have 2 variables), I want to use this one first.
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I want to solve this way, because it has x, and this has x--the same variable.
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And you want to stick to the same variable.
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Here I can say 5x (and I am going to use that, if consecutive interior angles are supplementary, then the lines are parallel) + 9x + 12 = 180.
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And then, if I do 14x + 12 = 180, 14x =...if I subtract that out, it is going to be 168.
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Then, x equals, let's see, 12; so if x is 12, then that is my value of x, and then I have to find the value of y.
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Now, this angle measure, then, if x is 12, will be 60, because 5 times 12 is 60.
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Then, this right here: 9 times 12, plus 12, is going to be 120.
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Now, I also know that this and this are supplementary, so the 60 + 120 has to be 180; that is one way to check your answer.
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Now, remember: earlier, we said that this angle and this angle are also consecutive interior angles, which means that they are supplementary.
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Well, if this is 60, then this has to be 120, because this angle and this angle are supplementary; they are consecutive interior angles.
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So, since this is 120, to solve for y, I can just make this whole thing equal to 120.
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7y + 10 = 120; subtract the 10; I get 7y = 110; and then, y = 110/7.
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And that is simplified as much as possible, so that would be the answer.
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Now, when you get a fraction, it is fine; it is OK if you get a fraction.
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You can leave it as an improper fraction, like that, or you can change it to a mixed number.
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But this is fine, whichever way you do it, as long as it is simplified
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(meaning there are no common factors between the top number and the bottom number, the numerator and denominator).
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Then, you are OK; there is x, and there is y.
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You solved for the x-value and the y-value, so now that these consecutive interior angles are supplementary, I can say that these sides are parallel.
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And then, since this one and this one, consecutive interior angles, are supplementary, this side and this side are parallel.
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Now, notice how, for these, I did it once; and then, for these, I had to do it twice,
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because any time you have the same number of these little marks, then you are saying that they are congruent;
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if they are slash marks, then they are congruent; if they are these marks, then they are parallel.
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If it is one time, then all of the lines with one will be parallel.
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For these, all of the lines with two will be parallel; if you have another pair of parallel lines, then you can do those three times.
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OK, the last example: we are going to do a proof.
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Write a two-column proof: before we begin, we should always look at what is given, what you have to prove, and the diagram.
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Look at it and see how you are going to get from point A to point B.
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This is what we are trying to prove: that AB is parallel to EF.
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Angle 1 and angle 2 are congruent--this is congruent; and then, angle 1 is also congruent to 3.
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So, one time is congruent...two times are congruent.
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We know that, since these two angles are congruent, and these two angles are congruent,
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I know that, since these two angles are going to have some kind of special relationship,
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my theorem and my postulate say that, if they have a special relationship, then the lines are parallel.
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So, I am going to just do this step-by-step: here are my statements; my reasons I will put right here.
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Statement #1: We know that we have to write the given, so it is that (let me write it a little bit higher; I am out of room)...
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#1 is that angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3.
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And the reason for this is that it is given.
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#2: Well, if angle 1 is congruent to angle 2, and angle 1 is congruent to angle 3, then I can say that angle 2 is congruent to angle 3.
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So, I will read it this way: If angle 2 is congruent to angle 1, and angle 1 is congruent to angle 3, then angle 2 is congruent to angle 3.
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And this is the transitive property of congruency--not equality, but congruency.
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#1: If a is equal to b, and b = c, then a = c; that is the transitive property, and we are using congruency, so it is not equality; it is congruency.
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Then, from there, since I proved that angle 2 is congruent to angle 3, I know that they are alternate interior angles.
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Alternate interior angles are congruent; that means that I can just say that, since alternate interior angles are congruent, then these lines are parallel.
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Step 3: You can say that angles 2 and 3 are labeled as alternate interior angles.
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Or, you can just go ahead and write out what the "prove" statement is--what you are trying to prove,
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since you already proved that they are congruent, and they are alternate interior angles.
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Depending on how your teacher wants you to set this up, this will be either step 3 or step 4.
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My reason is going to be: If alternate interior angles are congruent, then the lines...
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now, this is not the complete theorem, but you can just shorten it...are parallel.
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And that would be the proof; and make sure (again, since we haven't done proofs in a while)
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that the given statement always comes first, and the "prove" statement always comes last.
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It is like you are trying to get from point A to point B.
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If you are driving somewhere--you start from your house, and you are driving to school--your house is point A, and your school is point B.
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There are steps to get there; it is the same thing--proofs are exactly the same way.
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You need to have your steps to get from point A to point B.
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That is it for this lesson; we will see you soon; thanks for watching Educator.com.