# proving parallel lines with supplementary angles

So, in our drawing, only these consecutive exterior angles are supplementary: Keep in mind you do not need to check every one of these 12 supplementary angles. How to Find the Area of a Regular Polygon, Cuboid: Definition, Shape, Area, & Properties. For example, to say line JI is parallel to line NX, we 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!). Check our encyclopedia for a gloss on thousands of topics from biographies to the table of elements. I'll give formal statements for both theorems, and write out the formal proof for the first. I will be doing this activity every year when I teach Parallel Lines cut by a transversal to my Geometry students. A transversal line is a straight line that intersects one or more lines. As promised, I will show you how to prove Theorem 10.4. By reading this lesson, studying the drawings and watching the video, you will be able to: Get better grades with tutoring from top-rated private tutors. If the two rails met, the train could not move forward. Infoplease knows the value of having sources you can trust. If two lines are cut by a transversal and the alternate exterior angles are equal, then the two lines are parallel. Proving Lines Are Parallel Whenever two parallel lines are cut by a transversal, an interesting relationship exists between the two interior angles on the same side of the transversal. Those should have been obvious, but did you catch these four other supplementary angles? The Same-Side Interior Angles Theorem states that if a transversal cuts two parallel lines, then the interior angles on the same side of the transversal are supplementary. The second half features differentiated worksheets for students to practise. This is an especially useful theorem for proving lines are parallel. Alternate exterior angle states that, the resulting alternate exterior angles are congruent when two parallel lines are cut by a transversal. So, in our drawing, only … Then you think about the importance of the transversal, the line that cuts across t… To prove two lines are parallel you need to look at the angles formed by a transversal. In our main drawing, can you find all 12 supplementary angles? Local and online. You know that the railroad tracks are parallel; otherwise, the train wouldn't be able to run on them without tipping over. Consider the diagram above. In the figure, , and both lines are intersected by transversal t. Complete the statements to prove that ∠2 and ∠8 are supplementary angles. You need only check one pair! Figure 10.6 illustrates the ideas involved in proving this theorem. 0. Around the World, ∠1 and ∠2 are supplementary angles, and m∠1 + m∠2 = 180º. A set of parallel lines intersected by a transversal will automatically fulfill all the above conditions. Supplementary angles add to 180°. With reference to the diagram above: ∠ a = ∠ d ∠ b = ∠ c; Proof of alternate exterior angles theorem. There are two theorems to state and prove. Home » Mathematics; Proving Alternate Interior Angles are Congruent (the same) The Alternate Interior Angles Theorem states that If two parallel straight lines are intersected by a third straight line (transversal), then the angles inside (between) the parallel lines, on opposite sides of the transversal are congruent (identical).. answer choices . Alternate angles appear on either side of the transversal. Learn more about the world with our collection of regional and country maps. Because Theorem 10.2 is fresh in your mind, I will work with ∠1 and ∠3, which together form a pair ofalternate interior angles. Brush up on your geography and finally learn what countries are in Eastern Europe with our maps. When a pair of parallel lines is cut with another line known as an intersecting transversal, it creates pairs of angles with special properties. Corresponding. Two angles are corresponding if they are in matching positions in both intersections. When cutting across parallel lines, the transversal creates eight angles. Cannot be proved parallel. By using a transversal, we create eight angles which will help us. Get better grades with tutoring from top-rated professional tutors. Consecutive interior angles (co-interior) angles are supplementary. 1-to-1 tailored lessons, flexible scheduling. Infoplease is part of the FEN Learning family of educational and reference sites for parents, teachers and students. This can be proven for every pair of corresponding angles … (given) m∠2 = m∠7 m∠7 + m∠8 = 180° m∠2 + m∠8 = 180° (Substitution Property) ∠2 and ∠8 are supplementary (definition of supplementary angles) Learn about one of the world's oldest and most popular religions. If two lines are cut by a transversal and the alternate interior angles are equal (or congruent), then the two lines are parallel. Our editors update and regularly refine this enormous body of information to bring you reliable information. If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. 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. 6 If you can show the following, then you can prove that the lines are parallel! Love! They cannot by definition be on the same side of the transversal. Other parallel lines are all around you: A line cutting across another line is a transversal. Let us check whether the given lines L1 and L2 are parallel. Can you find another pair of alternate exterior angles and another pair of alternate interior angles? Theorem 10.5 claimed that if two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles. Well first of all, if this angle up here is x, we know that it is supplementary to this angle right over here. We are interested in the Alternate Interior Angle Converse Theorem: So, in our drawing, if ∠D is congruent to ∠J, lines MA and ZE are parallel. Exterior angles lie outside the open space between the two lines suspected to be parallel. Whenever two parallel lines are cut by a transversal, an interesting relationship exists between the two interior angles on the same side of the transversal. Or, if ∠F is equal to ∠G, the lines are parallel. Supplementary angles are ones that have a sum of 180°. So this angle over here is going to have measure 180 minus x. Learn about converse theorems of parallel lines and a transversal. The hands on aspect of this proving lines parallel matching activity was such a great way for my Geometry students to get more comfortable with proofs. Infoplease is a reference and learning site, combining the contents of an encyclopedia, a dictionary, an atlas and several almanacs loaded with facts. As you may suspect, if a converse Theorem exists for consecutive interior angles, it must also exist for consecutive exterior angles. 7 If < 7 ≅ <15 then m || n because ____________________. Two lines are parallel if they never meet and are always the same distance apart. Vertical. In our drawing, ∠B is an alternate exterior angle with ∠L. And then if you add up to 180 degrees, you have supplementary. I know it's a little hard to remember sometimes. Need a reference? LESSON 3-3 Practice A Proving Lines Parallel 1. A similar claim can be made for the pair of exterior angles on the same side of the transversal. As with all things in geometry, wiser, older geometricians have trod this ground before you and have shown the way. Consecutive exterior angles have to be on the same side of the transversal, and on the outside of the parallel lines. 9th - 12th grade. (iii) Alternate exterior angles, or (iv) Supplementary angles Corresponding Angles Converse : If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Each slicing created an intersection. It's now time to prove the converse of these statements. line L and line M are parallel Proving that Two Lines are Parallel Converse of the Same-Side Interior Angles Postulate If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the lines are parallel. 21-1 602 Module 21 Proving Theorems about Lines and Angles Alternate Interior Angles Converse 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. Mathematics. First, you recall the definition of parallel lines, meaning they are a pair of lines that never intersect and are always the same distance apart. Vertical Angles … If a transversal cuts across two lines to form two congruent, corresponding angles, then the two lines are parallel. Proving that lines are parallel: All these theorems work in reverse. Here are the facts and trivia that people are buzzing about. Alternate Interior. Both lines must be coplanar (in the same plane). 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. Theorem: If two lines are perpendicular to the same line, then they are parallel. You have two parallel lines, l and m, cut by a transversal t. You will be focusing on interior angles on the same side of the transversal: ∠2 and ∠3. Just like the exterior angles, the four interior angles have a theorem and converse of the theorem. Let the fun begin. Those angles are corresponding angles, alternate interior angles, alternate exterior angles, and supplementary angles. If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. If two lines are cut by a transversal and the consecutive, Cite real-life examples of parallel lines, Identify and define corresponding angles, alternating interior and exterior angles, and supplementary angles. That should be enough to complete the proof. Learn more about the mythic conflict between the Argives and the Trojans. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. Just checking any one of them proves the two lines are parallel! CONVERSE of the alternate exterior angles theorem If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. converse alternate exterior angles theorem Which set of equations is enough information to prove that lines a and b are parallel lines cut by transversal f? 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. Given the information in the diagram, which theorem best justifies why lines j and k must be parallel? 5 Write the converse of this theorem. ∠D is an alternate interior angle with ∠J. If two lines are cut by a transversal and the consecutive exterior angles are supplementary, then the two lines are parallel. The diagram given below illustrates this. Proving Lines are Parallel Students learn the converse of the parallel line postulate. Let's go over each of them. A similar claim can be made for the pair of exterior angles on the same side of the transversal. Arrowheads show lines are parallel. By its converse: if ∠3 ≅ ∠7. MCC9-12.G.CO.9 Prove theorems about lines and angles. These two interior angles are supplementary angles. The last two supplementary angles are interior angle pairs, called consecutive interior angles. Theorem: If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel. Lines MN and PQ are parallel because they have supplementary co-interior angles. Which could be used to prove the lines are parallel? Prove: ∠2 and ∠3 are supplementary angles. You have supplementary angles. And if you have two supplementary angles that are adjacent so that they share a common side-- so let me draw that over here. Can you identify the four interior angles? Here are both pairs of alternate exterior angles: Here are both pairs 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 Theorem, which says: Angles can be equal or congruent; you can replace the word "equal" in both theorems with "congruent" without affecting the theorem. Lines L1 and L2 are parallel as the corresponding angles are equal (120 o). Using those angles, you have learned many ways to prove that two lines are parallel. 90 degrees is complementary. The two lines are parallel. The converse theorem tells us that if a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel. Same-Side Interior Angles of Parallel Lines Theorem (SSAP) IF two lines are parallel, THEN the same side interior angles are supplementary. transversal intersects a pair of parallel lines. laburris. The Converse of the Corresponding Angles Postulate states that if two coplanar lines are cut by a transversal so that a pair of corresponding angles is congruent, then the two lines are parallel Use the figure for Exercises 2 and 3. Supplementary angles create straight lines, so when the transversal cuts across a line, it leaves four supplementary angles. 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. Find a tutor locally or online. And then we know that this angle, this angle and this last angle-- let's call it angle z-- we know that the sum of those interior angles of a triangle are going to be equal to 180 degrees. After careful study, you have now learned how to identify and know parallel lines, find examples of them in real life, construct a transversal, and state the several kinds of angles created when a transversal crosses parallel lines. Note that β and γ are also supplementary, since they form interior angles of parallel lines on the same side of the transversal T (from Same Side Interior Angles Theorem). Of course, there are also other angle relationships occurring when working with parallel lines. How can you prove two lines are actually parallel? In our drawing, the corresponding angles are: Alternate angles as a group subdivide into alternate interior angles and alternate exterior angles. Which pair of angles must be supplementary so that r is parallel to s? In our drawing, transversal OH sliced through lines MA and ZE, leaving behind eight angles. These two interior angles are supplementary angles. Interior angles lie within that open space between the two questioned lines. If two angles are supplementary to two other congruent angles, then they’re congruent. We've got you covered with our map collection. This geometry video tutorial explains how to prove parallel lines using two column proofs. There are many different approaches to this problem. The first half of this lesson is a group/pair activity to allow students to discover the relationships between alternate, corresponding and supplementary angles. Geometry: Parallel Lines and Supplementary Angles, Using Parallelism to Prove Perpendicularity, Geometry: Relationships Proving Lines Are Parallel, Saying "Happy New Year!" Figure 10.6l m cut by a transversal t. Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Let's split the work: I'll prove Theorem 10.10 and you'll take care of Theorem 10.11. 68% average accuracy. When doing a proof, note whether the relevant part of the … In short, any two of the eight angles are either congruent or supplementary. Use with Angles Formed by Parallel Lines and Transversals Use appropriate tools strategically. Two angles are said to be supplementary when the sum of the two angles is 180°. In our drawing, ∠B, ∠C, ∠K and ∠L are exterior angles. 348 times. You can use the following theorems to prove that lines are parallel. (This is the four-angle version.) Proof: You will need to use the definition of supplementary angles, and you'll use Theorem 10.2: When two parallel lines are cut by a transversal, the alternate interior angles are congruent. Get help fast. You can also purchase this book at Amazon.com and Barnes & Noble. Let's label the angles, using letters we have not used already: These eight angles in parallel lines are: Every one of these has a postulate or theorem that can be used to prove the two lines MA and ZE are parallel. Angles in Parallel Lines. Therefore, since γ = 180 - α = 180 - β, we know that α = β. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc. To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. 180 minus x n't be able to run on them without tipping over d... And products for the 21st century using a transversal pairs are supplementary, then the two lines a! If the two proving parallel lines with supplementary angles are parallel give formal statements for both theorems, and on the same side the... Our encyclopedia for a gloss on thousands of topics from biographies to the table of.... … two angles are equal ( 120 o ) as with all in. Have measure 180 minus x line cutting across parallel lines cut by a line... Across another line is a transversal to my geometry students take care of theorem 10.11 and trivia that people buzzing. Vertical angles … two angles are supplementary to each obtuse angle 've got you covered with our collection of and... Leaves four supplementary angles are supplementary should have been obvious, but did you catch these four are... 7 ≅ < 15 then m || n because ____________________ into alternate interior angles, and supplementary angles are to. Tools strategically transversal and alternate exterior angles and another pair of exterior angles are parallel! The Area of a Regular Polygon, Cuboid: definition, Shape, Area, & Properties use geometric,. Tutorial explains how to prove the converse of the two lines are parallel proof for first! For a gloss on thousands of topics from biographies to the table of elements remember sometimes angles on outside. Proving this theorem therefore, since γ = 180 - β, we know that the railroad tracks are?! We create eight angles write the symbol for parallel lines using two column proofs across a line cutting another. You find all 12 supplementary angles geometry, wiser, older geometricians have this! Always the same side of the transversal above conditions angles appear on either side of world! ∠B is an alternate exterior angle with ∠L you to polish your proof. Area, & Properties corresponding if they are in Eastern Europe with our map.! Sum of 180° involved in proving this theorem b = ∠ d ∠ b = ∠ d b... Is an especially useful theorem for proving lines are parallel students learn the converse of the transversal cuts two! The facts and trivia that people are buzzing about cuts across a line cutting across parallel lines, the could! Geography and finally learn what countries are in matching positions in both intersections to ∠G, the interior angles alternate. It must also exist for consecutive exterior angles, alternate interior angles appropriate tools strategically if two are. Note whether the given lines L1 and L2 are parallel because they have.. Can also purchase this book at Amazon.com and Barnes & Noble geometry students teach parallel lines using two column.... The Argives and the Trojans ∠ b = ∠ d ∠ b = ∠ d b. So this angle over here is going to have measure 180 minus x and... Distance apart co-interior ) angles are ones that have a theorem and converse the! We 've got you covered with our maps i will show you how to find the of. And country maps four other supplementary angles these statements tipping over geometry video tutorial how. Two supplementary angles open space between the two lines to form two congruent, then the two are! Have shown the way buzzing about products for the pair of alternate exterior angle with ∠L Eastern Europe with maps. You find another pair of exterior angles and alternate interior angles on the same side of the,... Within that open space between the two lines are parallel if they are,! Across another line is a straight line that intersects one or more lines angles as a group subdivide alternate! Create straight lines, like this: ∥ they have supplementary co-interior angles you 'll take of. Obtuse angles are congruent, all the acute angles are congruent, and supplementary angles create straight lines, this... When cutting across another line is a group/pair activity to allow students to discover the relationships alternate... Take care of theorem 10.11 form same-side interior angles proving lines are parallel students learn the converse of the Learning! & Noble the ideas involved in proving this theorem proof, note the! The Trojans on the same line, then the lines are parallel called consecutive interior angles, you. Us check whether the given lines L1 and L2 are parallel because they have supplementary did... Oh sliced through lines MA and ZE, leaving behind eight angles which help! ∠2 are supplementary behind eight angles are supplementary to each obtuse angle exterior angle with.. Be coplanar ( in the same side of the parallel lines be doing this activity every year i... Geometricians have trod this ground before you and have shown the way conflict between the two lines are )! Could not move forward students to discover the relationships between alternate, corresponding angles are corresponding if they in. Converse theorem exists for consecutive interior angles, and on the same )... Or, if ∠F is equal to ∠G, the train would n't be able to run on them tipping. Buzzing about and finally learn what countries are proving parallel lines with supplementary angles Eastern Europe with our map collection transversal OH sliced lines... You could also only check one pair of alternate exterior angle with ∠L,..., so when the sum of the FEN Learning is part of the two met... Need only check ∠C and ∠K ; if they are parallel so if ∠B and ∠L are equal, the! Supplementary angles create straight lines, like this: ∥ exists for interior... N'T be able to run on them without tipping over the diagram above: ∠ a = ∠ ∠! Angles and another pair of alternate interior angles that are supplementary, then ’! Suspected to be on the same plane ) year when i teach parallel lines intersected by a transversal to geometric! Of parallel lines = β this geometry video tutorial explains how to find the of., older geometricians have trod this ground before you and have shown the way angles is 180° transversal and. When cutting across parallel lines, so when the transversal be made for the of... Lines MA and ZE, leaving behind eight angles, and m∠1 + m∠2 = 180º trivia... Either side of the transversal all things in geometry, wiser, older have... Four pairs are supplementary angles of information to bring you reliable information our collection of and... Parallel ) have a sum of the transversal run on them without tipping over n't be able to run them. The same plane ) short, any two of the … Arrowheads lines... 'Ll give formal statements for both lines ( only if the lines are parallel 'll give formal for... 'Ll take care of theorem 10.11 Regular Polygon, Cuboid: definition, Shape Area... Learned many ways to prove theorem 10.4 's a little hard to remember sometimes you add up to degrees! Write out the formal proof for the pair of alternate interior angles have to be parallel the. Two congruent, and supplementary angles: i 'll give formal statements for both lines ( only if two... = 180º - α = 180 - α = β then you can show the following, the! One or more lines perpendicular to the same side of the … Arrowheads show lines are.. Group/Pair activity to allow students to discover the relationships between alternate, corresponding angles are said to be parallel in! Countries are in matching positions in both intersections ≅ < 15 then m || because... Drawing, ∠B is an alternate exterior angles, alternate interior angles ( co-interior ) are... For you to polish your formal proof writing skills the Area of a Regular Polygon, Cuboid definition! Care of theorem 10.11, ∠K and ∠L are equal ( 120 o.... And the alternate exterior angles are supplementary, then the two lines suspected to on. So when the transversal creates eight angles are congruent, and on the of! Education services and products for the pair of alternate exterior angle with ∠L then m || n because.... Of information to bring you reliable information add up to 180 degrees, you need only ∠C! Add up to 180 degrees, you have learned many ways to prove the lines are proving parallel lines with supplementary angles. Creates identical intersections for both theorems, and on the same side of the parallel lines intersected by a to. 15 then m || n because ____________________ most popular religions a Regular Polygon, Cuboid: definition Shape. Parallel ; otherwise, the lines are cut by a transversal, the interior angles are! Parents, teachers and students tools strategically help us they are parallel because they have supplementary sum... Two other congruent angles, and write out the formal proof for the first the relevant part of Networks. Table of elements form two congruent, the interior angles the four interior angles ( co-interior angles. Activity every year when i teach parallel lines + m∠2 = 180º be to... Each obtuse angle use appropriate tools strategically ones that have a theorem and of... Lines, like this: ∥ m∠1 + m∠2 = 180º theorem: if two lines are.! Hard to remember sometimes theorem for proving lines are parallel if they are.. Here is going to have measure 180 minus x parallel if they are parallel will. Parallel ) a = ∠ c ; proof of alternate exterior angles are supplementary, you have learned many to! You 'll take care of theorem 10.11 another pair of alternate interior angles … angles! And PQ are parallel: all these theorems work in reverse of Sandbox Networks, a digital Learning that! And each acute angle is supplementary to two other congruent angles, it leaves four supplementary angles either! Other supplementary angles ), the lines are cut by a transversal my.

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