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Proving Parallel Lines
Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript We know that if we have two lines that are parallel-- so let me draw those two parallel lines, l and m. So that's line l and line m.
We know that if they are parallel, then if we were to draw a transversal that intersects both of them, that the corresponding angles are equal. So this is x, and this is y So we know that if l is parallel to m, then x is equal to y. What I want to do in this video is prove it the other way around. I want to prove-- So this is what we know. We know this. What I want to do is prove if x is equal to y, then l is parallel to m. So that we can go either way. If they're parallel, then the corresponding angles are equal.
And I want to show if the corresponding angles are equal, then the lines are definitely parallel. And what I'm going to do is prove it by contradiction.Parallel lines are important when you study quadrilaterals because six of the seven types of quadrilaterals all of them except the kite contain parallel lines.
The eight angles formed by parallel lines and a transversal are either congruent or supplementary. Check out the above figure which shows three lines that kind of resemble a giant not-equal sign.
The two horizontal lines are parallel, and the third line that crosses them is called a transversal. As you can see, the three lines form eight angles. Proving that angles are congruent: If a transversal intersects two parallel lines, then the following angles are congruent refer to the above figure :.
Alternate interior angles: The pair of angles 3 and 6 as well as 4 and 5 are alternate interior angles. These angle pairs are on opposite alternate sides of the transversal and are in between in the interior of the parallel lines. Alternate exterior angles: Angles 1 and 8 and angles 2 and 7 are called alternate exterior angles. Corresponding angles: The pair of angles 1 and 5 also 2 and 6, 3 and 7, and 4 and 8 are corresponding angles. Angles 1 and 5 are corresponding because each is in the same position the upper left-hand corner in its group of four angles.
Also notice that angles 1 and 4, 2 and 3, 5 and 8, and 6 and 7 are across from each other, forming vertical angles, which are also congruent. Proving that angles are supplementary: If a transversal intersects two parallel lines, then the following angles are supplementary see the above figure :.
You can sum up the above definitions and theorems with the following simple, concise idea. When you have two parallel lines cut by a transversal, you get four acute angles and four obtuse angles except when you get eight right angles.
All the acute angles are congruent, all the obtuse angles are congruent, and each acute angle is supplementary to each obtuse angle. In short, any two of the eight angles are either congruent or supplementary. Proving that lines are parallel: All these theorems work in reverse. You can use the following theorems to prove that lines are parallel.
Definitions and Theorems of Parallel Lines.How can you prove two lines are actually parallel? As with all things in geometry, wiser, older geometricians have trod this ground before you and have shown the way.
By using a transversal, we create eight angles which will help us. Two lines are parallel if they never meet and are always the same distance apart. Both lines must be coplanar in the same plane.
For example, to say line J I is parallel to line N Xwe write:. If you have ever stood on unused railroad tracks and wondered why they seem to meet at a point far away, you have experienced parallel lines and perspective! If the two rails met, the train could not move forward. Other parallel lines are all around you:.
A line cutting across another line is a transversal. When cutting across parallel lines, the transversal creates eight angles. Create a transversal using any existing pair of parallel lines, by using a straightedge to draw a transversal across the two lines, like this:.
Those eight angles can be sorted out into pairs. Let's label the angles, using letters we have not used already:. Every one of these has a postulate or theorem that can be used to prove the two lines M A and Z E are parallel.Geometry: Section 3.3- Proving Lines are Parallel
Let's go over each of them. The Corresponding Angles Postulate states that parallel lines cut by a transversal yield congruent corresponding angles.
Angle relationships with parallel lines
We want the converse of that, or the same idea the other way around:. To know if we have two corresponding angles that are congruent, we need to know what corresponding angles are.
Each slicing created an intersection. If one angle at one intersection is the same as another angle in the same position in the other intersection, then the two lines must be parallel.
Angles, Parallel Lines, & Transversals
Two angles are corresponding if they are in matching positions in both intersections. Alternate angles as a group subdivide into alternate interior angles and alternate exterior angles. Exterior angles lie outside the open space between the two lines suspected to be parallel. Interior angles lie within that open space between the two questioned lines. Can you identify the four interior angles? Alternate angles appear on either side of the transversal.
They cannot by definition be on the same side of the transversal. Can you find another pair of alternate exterior angles and another pair of alternate interior angles? If just one of our two pairs of alternate exterior angles are equal, then the two lines are parallel, because of the Alternate Exterior Angle Converse Theoremwhich says:.
Angles can be equal or congruent ; you can replace the word "equal" in both theorems with "congruent" without affecting the theorem.
You need only check one pair! Just like the exterior angles, the four interior angles have a theorem and converse of the theorem. Again, you need only check one pair of alternate interior angles!
Supplementary angles create straight lines, so when the transversal cuts across a line, it leaves four supplementary angles. When a transversal cuts across lines suspected of being parallel, you might think it only creates eight supplementary angles, because you doubled the number of lines. Those should have been obvious, but did you catch these four other supplementary angles?
These four pairs are supplementary because the transversal creates identical intersections for both lines only if the lines are parallel.Played 58 times. Print Share Edit Delete.
Live Game Live. Finish Editing. This quiz is incomplete! To play this quiz, please finish editing it. Delete Quiz. Question 1. Which pair of angles must be supplementary so that r is parallel to s?
Which value of x will show that lines l and m are parallel? What value of x proves that line p is parallel to line r? Which value of x will show that line m is parallel to line r? What value of x proves that lines l and m are parallel? Which of the following relationship proves that j is parallel to k? Line v is a transversal. Which is a true statement? Which condition will prove that line l is parallel to line m?
Line l and m are cut by transversal n. Which statement proves that line l is parallel to line m? Are lines j and k skew? Are lines m and n skew? Are lines u and w skew? Quizzes you may like. Proving Parallel Lines. Parallel, Perpendicular, or Neither?? Parallel and Perpendicular Lines. Angles - Parallel Lines Rules! Find a quiz All quizzes. All quizzes.
Create a new quiz. Find a quiz Create a quiz My quizzes Reports Classes new.If two angles are vertical angles, then they are congruent. You learned this as the Vertical Angles Theorem. Can you reverse this statement? The converse of a logical statement is made by reversing the hypothesis and the conclusion in an if-then statement.
In this case, no. There are many examples of congruent angles that are not vertical angles—for example the corners of a square. Sometimes the converse of an if-then statement will also be true. Can you think of an example of a statement in which the converse is true? This lesson explores converses to the postulates and theorems about parallel lines and transversals. Previously you learned that "if two parallel lines are cut by a transversal, the corresponding angles will be congruent.
If corresponding angles are congruent when two lines are crossed by a transversal, then the two lines crossed by the transversal are parallel. Suppose we know that and. What can we conclude about lines and? Notice that and are corresponding angles.
Sincewe can apply the Converse of the Corresponding Angles Postulate and conclude that. You can also use converse statements in combination with more complex logical reasoning to prove whether lines are parallel in real life contexts. The following example shows a use of the contrapositive of the Corresponding Angles Postulate. The three lines in the figure below represent metal bars and a cable supporting a water tower. Are the lines and parallel?
To find out whether lines and are parallel, you must identify the corresponding angles and see if they are congruent. In this diagram, and are corresponding angles because they are formed by the transversal and the two lines crossed by the transversal and they are in the same relative place.
The problem states that and. Thus, they are not congruent. If those two angles are not congruent, the lines are not parallel. In this scenario, the lines and and thus the support bars they represent are NOT parallel. Note that just because two lines may look parallel in the picture that is not enough information to say that the lines are parallel. To prove two lines are parallel you need to look at the angles formed by a transversal.
Another important theorem you derived in the last lesson was that when parallel lines are cut by a transversal, the alternate interior angles formed will be congruent. If two lines are crossed by a transversal and alternate interior angles are congruent, then the lines are parallel. Given and are crossed by and. Notice in the proof that we had to show that the corresponding angles were congruent.
Once we had done that, we satisfied the conditions of the Converse of the Corresponding Angles postulate, and we could use that in the final step to prove that the lines are parallel.When the students arrive, there is a sentence displayed on the board. I collect their homework and ask that they direct their attention to the sentence: If I leave the house, then I must have my cell phone on me.
I ask the students to identify the hypothesis and the conclusion of this conditionalas well as the truth valueand then ask the students to provide a few more examples of conditional statements, for which we again identify the hypotheses and conclusions, and discuss the truth values MP2. Then I introduce the concepts of inverse, converse, and contrapositive using the statement If p, then q. I ask that the students create the inverse, converse and the contrapositive of the cell phone statement I originally gave them, and I raise the issue of the truth values of these statements.
If the original statement is true, do the other statements have to be true? In order to help my students understand the possibilities with regard to truth values, I use a Venn diagram, as I demonstrate in this video. Our ultimate conclusion is that the original statement and the contrapositive will always have the same truth value; the inverse and the converse only sometimes share that truth value.
I really miss this unit my students always seemed to enjoy it! When talking about the isosceles triangle theorems, for example, the use of the word "converse" is really helpful, and serves, I think, to solidify the students' understanding of these theorems. I remind the students that we began our study of angles and parallel lines with this postulate: If two parallel lines are cut by a transversal, the corresponding angles are congruent. I ask them to tell me what the converse of this statement might be, and explain that we will postulate the converse as well.
From this postulate, we then prove the theorem " If two parallel lines are cut by a transversal, the alternate interior angles are congruent. I provide the students with the hand out, Proving Lines are Paralleland work through these proofs with the class.
In the second of the proofs, the students can choose which angles pairs they use in their proof, so that some can use alternate interior angles, while others can use corresponding angles. We discuss both options. In this lesson, I have chosen to do two-column proofs. In future lessons, we will also use flow chart proofs and paragraph proofs. I think it's important for students to be exposed to the different approaches to proof, and believe they should be able to choose the format that is most natural for them.
This line needs to be parallel to the one next to it. How can we use our knowledge of parallel lines to ensure that the two lines will be parallel?
This question leads us into the construction of a line parallel to a given line through a given point. On the back of the sheet the plain side I introduce the students to the construction for copying an angle.
Then we turn to the front of the sheet and I instruct the students on the parallel line construction. When we have completed copying the angle and drawing in the parallel line, I ask the students to tell me what type of angle pair we just constructed and the theorem that we used to guarantee that our lines are parallel.
We now change gears entirely and focus on algebra. With a 90 minute class, I think it's crucial to change activities and focus often.
Otherwise the class could get awfully boring! I announce that we are going to refresh our memories of algebra, and write two equations with two unknowns on the board that can be easily solved using substitution. I ask the students to solve for x and y, and we discuss their answers and their approach. We discuss this method, and I give the students several problems to work through. Now it's time to apply our algebra skills to parallel lines.In this lesson we are 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.
We learned, from the corresponding angles postulate, that if the lines are parallel, then the corresponding angles are congruent. If the lines are the parallel lines that are cut by a transversal, then the corresponding angles are congruent.
We'll use these and some others in our proofs. Get immediate access to our entire library. Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. 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.
We learned, from the Corresponding Angles Postulate, that if the lines are parallel, then the corresponding angles are congruent. 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.
So, it is using the converse of that postulate that we learned a couple of lessons ago. Before, when we used that postulate, it was given that the lines were parallel; and then you would have to show But for this one, they are giving you that the corresponding angles are congruent. And then, the conclusion, "the lines are parallel," is what you are going to be proving. This first postulate: if you look at angles 1 and 2, those are corresponding angles.
So, if I tell you that angle 1 and angle 2 are congruent, and they are corresponding angles, then the lines are parallel. As long as these two angles are congruent, then these lines are parallel; so I can conclude that these lines are parallel. Now, again, for them to be corresponding angles, the lines don't have to be parallel. Even if the lines are not parallel, they are still considered corresponding angles. But now, we know that, as long as the corresponding angles are congruent, then the lines will be parallel.
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.
The alternate exterior angles theorem said that, if two lines are parallel, then alternate exterior angles are congruent. This is the converse; so they are giving you that alternate exterior angles are congruent; then, the lines are parallel.