Show 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. These two interior angles are supplementary angles. A similar claim can be made for the pair of exterior angles on the same side of the transversal. There are two theorems to state and prove. I'll give formal statements for both theorems, and write out the formal proof for the first. The second theorem will provide yet another opportunity for you to polish your formal proof writing skills.
Let the fun begin. As promised, I will show you how to prove Theorem 10.4. Figure 10.6 illustrates the ideas involved in proving this theorem. 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. You'll need to relate to one of these angles using one of the following: corresponding angles, vertical angles, or alternate interior angles. There are many different approaches to this problem. Because Theorem 10.2 is fresh in your mind, I will work with 1 and 3, which together form a pair ofalternate interior angles.
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. 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. You can also purchase this book at Amazon.com and Barnes & Noble.
In Geometry, we might have come across different types of lines, such as parallel lines, perpendicular lines, intersecting lines, and so on. Apart from that, we have another line called a transversal. This can be observed when a road crosses two or more roads, or a railway line crosses several other lines. These give a basic idea of a transversal. Transversals play an important role in establishing whether two or more other lines in the Euclidean plane are parallel. In this article, you will learn the definition of transversal line, angles made by the transversal with parallel and non-parallel lines with an example. Also, learn more concepts of geometry here. Table of Contents: Transversal MeaningA transversal is defined as a line that passes through two lines in the same plane at two distinct points in the geometry. A transversal intersection with two lines produces various types of angles in pairs, such as consecutive interior angles, corresponding angles and alternate angles. A transversal produces 8 angles and this can be observed from the figure given below: Transversal Line Definition A line that intersects two or more lines at distinct points is called a transversal line. Transversal LinesA transversal line can be obtained by intersecting two or more lines in a plane that may be parallel or non-parallel. In the above figure, line “t” is the transversal of two non-parallel lines a and b. Also, we can draw two transverse lines for two parallel lines and non-perpendicular lines. Transversal and Parallel LinesIn the below-given figure, the line RS represents the Transversal of Parallel Lines EF and GH. Transversal AnglesThe angles made by a transversal can be categorized into several types such as interior angles, exterior angles, pairs of corresponding angles, pairs of alternate interior angles, pairs of alternate exterior angles, and the pairs of interior angles on the same side of the transversal. All these angles can be identified in both cases, which means parallel and non-parallel lines. Transversal Lines and AnglesLet’s understand the angles made by transversal lines with other lines from the below table.
From the above table, we can say that the angles associated with the transversal will be the same in pairs in parallel and non-parallel lines. Click here to learn more about lines and angles. Transversal PropertiesSome of the properties of transversal lines with respect to the parallel lines are listed below.
LM is the transversal made by the parallel lines PQ and RS such that: The pair of corresponding angles that are represented with the same letters are equal.
Here, ∠3 + ∠5 = 180° and ∠4 + ∠6 = 180° Transversal ExampleLet’s have a look at the solved example given below: Question: In the given figure, AB and CD are parallel lines intersected by a transversal PQ at L and M, respectively. If ∠CMQ = 60°, find all other angles. Solution: A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles. Thus, the corresponding angles are equal. ∠ALM = ∠CMQ = 60° {given} We know that vertically opposite angles are equal. ∠LMD = ∠CMQ = 60° {given} And ∠ALM = ∠PLB = 60° Here, ∠CMQ + ∠QMD = 180° are the linear pair On rearranging, we get, ∠QMD = 180° – 60° = 120° Also, the corresponding angles are equal. ∠QMD = ∠MLB = 120° Now, ∠QMD = ∠CML = 120° {vertically opposite angles} ∠MLB = ∠ALP = 120° {vertically opposite angles}
When a transversal is formed by intersecting two parallel lines, then the following properties can be defined. 1. If a transversal cuts two parallel lines, each pair of corresponding angles are equal in measure. 2. If a transversal cuts two parallel lines, each pair of alternate interior angles are equal. 3. If a transversal cuts two parallel lines, then each pair of interior angles on the same side of the transversal are supplementary.
In geometry, a transversal is a line that intersects two or more lines at distinct points.
Various theorems are defined for transversal, such as: 1. If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. 2. If a transversal intersects two lines such that a pair of corresponding angles are equal, then the two lines are parallel to each other. 3. If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel. 4. If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
We can quickly identify the transversal since it crosses two or more lines at different points.
We do not have any particular symbol to represent a transversal in Maths.
We can label a transversal similar to other lines in geometry, which means using the English alphabet. For example, line PQ is the transversal of two lines AB and CD.
The symbol that denotes the similarity is ~. This can be read as “is similar to”. |