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In this video, we will learn how to use angle relationships formed by parallel lines and transversals to solve problems involving algebraic expressions and equations.
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First of all, letβs remember some properties of parallel lines and transversals.
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First of all, when any two lines intersect, we know that the vertically opposite angles will be congruent to one another.
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Opposite angles are two angles between secant lines that share a vertex, like we see here.
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The measure of angle π΄ would be equal to the measure of angle πΆ, and the measure of angle π΅ would be equal to the measure of angle π·.
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When any two secant lines intersect, we can also say that the adjacent angles will sum to 180 degrees.
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In this case, the measure of angle π΄ plus the measure of angle π· must be equal to 180 degrees.
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We could also say the measure of angle π· plus the measure of angle πΆ must be 180 degrees.
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In fact, any pair of adjacent angles in this image would sum to 180 degrees.
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But now we want to extend this to see what happens when two parallel lines are cut by the same transversal.
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Here, line two and line three are parallel, and theyβre both being cut by the transversal line one.
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If weβre just looking at line one and line two, weβve already shown that the vertically opposite angles are congruent to one another.
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But when we extend that to parallel lines cut by a transversal, we say if two or more parallel lines are cut by the same transversal, corresponding angles are congruent.
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Looking at the intersection of line one and line two, we could label the angles created as position one, two, three, and four.
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And then, since we know that line three is parallel to line two and is also being intersected by line one, when we label them in the same direction β top left, top right, bottom right, bottom left β the corresponding angles will be congruent, which means angles in position one are congruent to one another, angles in position two are congruent to one another, and so on.
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There are a few other things we should note.
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When two parallel lines are cut by a transversal, alternate angles are congruent.
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These are angles on opposite sides of the transversal.
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The one shown here are alternate interior angles, as they are on either side of the transversal but in between the two parallel lines.
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When two parallel lines are intersected by a transversal, there are two pairs of alternate interior angles.
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There are also two pairs of alternate exterior angles that are congruent.
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These are on either side of the transversal and on the outsides of the two parallel lines.
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Letβs look at one final angle pair relationship.
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Consecutive interior angles, or sometimes called cointerior angles, are angles on the same side of the transversal and in between the two parallel lines.
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These angles sum to 180 degrees.
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Itβs also worth noting that all of those properties extend to multiple parallel lines, not just two.
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In this image, we have three parallel lines, and for all three of the intersections, the corresponding angles will be congruent.
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But what if we have an additional transversal?
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For this other transversal, itβs of course true that the corresponding angles will be congruent.
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However, just because we know something about the angles of one of the transversals does not mean we can say anything about the angle relationships in the other transversal.
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So we have to remember to apply these properties to one transversal at a time.
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Letβs look at some examples.
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Work out the value of π₯ in the figure.
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First of all, letβs think about the type of lines weβre seeing in the figure.
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We have two parallel lines that are cut by a transversal, which makes the 61-degree angle and the π₯-angle alternate interior angles.
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And when two parallel lines are cut by a transversal, alternate interior angles are congruent.
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This means both of these angles are equal in measure and π₯ must be equal to 61.
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We can look at another example with a different set of angle pairs.
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Work out the value of π₯ in the figure.
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First of all, letβs think about what we can say about these lines.
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We have two parallel lines cut by a transversal, and here are the two angles weβre considering.
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Theyβre on the same side of the transversal and theyβre in the same position at either intersection, which means we call them corresponding angles.
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And when two parallel lines are cut by a transversal, corresponding angles are congruent, and therefore π₯ must be equal to 72.
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In our next example, weβll not only have two parallel lines, but weβll also have two transversals.
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Answer the questions for the given figure.
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Find the value of π₯ and find the value of π¦.
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First of all, letβs think about what we know about these lines.
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We have two parallel lines.
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Letβs call them line one and line two.
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And then we have a transversal, line three, that intersects both line one and line two.
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And we have a second transversal we can call line four that also intersects line one and line two.
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First, letβs just consider line one, two, and three, the left side of the image.
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Since line three is a transversal that cuts line one and line two, the angle π₯ and the angle 60 degrees are consecutive interior angles, sometimes called cointerior angles.
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And we know that these angles must sum to 180 degrees, which means π₯ plus 60 must be equal to 180.
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If we subtract 60 from both sides of this equation, we see that π₯ must be equal to 120.
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To solve for π¦, we can look at line one, two, and four.
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We have two parallel lines cut by a transversal, and the angle pair relationship of π¦ and 110 would be corresponding angles.
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Theyβre on the same side of the transversal and on the same side of their parallel lines, respectively.
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Theyβre in the same position at both intersections, and corresponding angles are congruent to one another, which means π¦ must be equal to 110.
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Now that we have all of this information, it would be possible to find this fourth angle inside this quadrilateral.
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This angle is a consecutive interior angle with 110 degrees, which means 110 plus this angle must equal 180.
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70 plus 110 equals 180.
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If we wanted to perform one final check that weβve calculated everything correctly, we could sum the four interior angles created by these lines.
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We know that the four interior angles inside a quadrilateral must sum to 360 degrees.
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In this case, they do, and that confirms that π₯ equals 120 and π¦ equals 110.
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Weβre ready to look at another example.
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The given figure shows a pair of parallel lines and two transversals, one of which crosses at right angles.
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Write an expression for π in terms of π.
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Using this expression for π, find a fully simplified expression for π in terms of π.
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Letβs see what we have.
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We have two parallel lines.
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We could call them line one and line two, and then we have two transversals, which we could call line three and line four.
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The first thing we wanna do is write an expression for π in terms of π.
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π and π occur at the intersection of line one and line three.
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They are adjacent angles on the same side of line one.
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This makes them supplementary angles.
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They must add up to 180 degrees.
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If we know that π degrees plus π degrees must equal 180 degrees and we want π in terms of π, this means we want to try to get π by itself and have π on the other side of the equation.
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To do that, we can subtract π degrees from both sides of the equation, which means π degrees will be equal to 180 degrees minus π degrees, and therefore π will be equal to 180 minus π.
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This completes part one, our first expression.
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Now, using this expression for π, we need to find a fully simplified expression for π in terms of π, which means we first need to look at the relationship between π degrees and π degrees.
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When it comes to π and π, they are in between the two parallel lines and on the same side of the transversal.
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They are consecutive interior angles, sometimes called cointerior angles.
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And when two parallel lines are crossed by a transversal, the consecutive interior angles are supplementary.
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They sum to 180.
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And that means π degrees plus π degrees must equal 180 degrees.
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For this expression, we want to express π in terms of π, and that means weβll need to get π by itself.
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First, we subtract π from both sides, which gives us π degrees equals 180 degrees minus π degrees or π equals 180 minus π.
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But remember, we want to substitute our expression weβve already found for π.
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We know that π equals 180 minus π.
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Remember though, weβre subtracting all of π, so we need to keep that in parentheses and then distribute that subtraction, which means subtract 180.
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But subtracting negative π would be adding π, and that means π equals 180 minus 180 plus π.
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180 minus 180 equals zero, and zero plus π equals π.
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A fully simplified expression for π in terms of π is that π is equal to π.
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In fact, if we take a closer look at π and π, we see that the relationship between these two angles are alternate interior angles, and we know that alternate interior angles will be congruent.
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So π in terms of π would be π equals 180 minus π, and π in terms of π would be π equals π.
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Weβre now ready to look at our final example.
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In the following figure, π§ equals two π₯ minus 69 and π€ equals two π¦ minus 59.
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Find π₯ and π¦.
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First of all, letβs see what we have.
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We have two parallel lines.
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If we extend them a little further, we can call them line one and line two.
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And then we have two transversals, which we can call line three and line four.
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At first, it seems like we have a lot of variables and not as much information.
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But when we look at the relationship of π€ degrees and π§ degrees, we can say that π§ degrees plus π€ degrees must equal 180 degrees.
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In addition to that, line three cuts through line one and line two.
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And that means this fourth interior angle would also be equal to π§ degrees since these two angles are corresponding.
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And this means our two transversals and our two parallel lines have given us a quadrilateral.
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And the interior angles of a quadrilateral must sum to 360 degrees.
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So we can say that π§ degrees plus π€ degrees plus 83 degrees plus π₯ must equal 360 degrees.
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But we already know what π§ plus π€ equals.
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Since π§ plus π€ equals 180 degrees, we can say 180 plus 83 plus π₯ must equal 360.
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So 263 degrees plus π₯ degrees equals 360 degrees.
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If we subtract 263 degrees from both sides, π₯ degrees equals 97 degrees and π₯ must equal 97.
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And because we know that π§ equals two π₯ minus 69, we can plug in 97 for π₯ and we find that π§ equals 125.
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Putting π§ equals 125 in our figure and π₯ equals 97, we can use this information to find π€.
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We know that π§ plus π€ equals 180.
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We plug in 125.
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And when we subtract 125 from both sides, we find out π€ equals 55.
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But remember, our goal is to find π₯ and π¦, and that means we need to plug in what we know about π€ to find π¦.
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Since π€ equals two π¦ minus 59, we can say 55 equals two π¦ minus 59.
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Add 59 to both sides, and we get 114 equals two π¦.
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Divide both sides by two; 114 divided by two is 57.
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So we can say that here π¦ must equal 57.
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In this figure under these conditions, π₯ is 97 and π¦ is 57.
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Before we finish, letβs review our key points.
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When two or more parallel lines are cut by a transversal, corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles or cointerior angles are supplementary.