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Perpendicular Lines Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2015 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: November 15, 2015 AUTHORS Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner www.ck12.org C HAPTER Chapter 1. Perpendicular Lines 1 Perpendicular Lines Here you’ll learn what perpendicular lines are and how to apply some basic properties and theorems about such lines. What if you were given a pair of lines that intersect each other at a 90◦ angle? What terminology would you use to describe such lines? After completing this Concept, you will be able to define perpendicular lines. You’ll also be able to apply the properties associated with such lines to solve for unknown angles. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136581 CK-12 Perpendicular Lines Watch the portions of this video dealing with perpendicular lines. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/1341 James Sousa: Perpendicular Lines Then watch this video. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/1344 James Sousa: Perpendicular Line Postulate Guidance Two lines are perpendicular when they intersect to form a 90◦ angle. Below, l ⊥ AB. 1 www.ck12.org In the definition of perpendicular the word “line” is used. However, line segments, rays and planes can also be perpendicular. The image below shows two parallel planes, with a third blue plane that is perpendicular to both of them. Basic Facts about Perpendicular Lines Theorem #1: If l||m and n ⊥ l, then n ⊥ m. Theorem #2: If l ⊥ n and n ⊥ m, then l||m. Postulate: For any line and a point not on the line, there is one line perpendicular to this line passing through the point. There are infinitely many lines that pass through A, but only one that is perpendicular to l. 2 www.ck12.org Chapter 1. Perpendicular Lines Example A Which of the following is the best example of perpendicular lines? 1. 2. 3. 4. Latitude on a Globe Opposite Sides of a Picture Frame Fence Posts Adjacent Sides of a Picture Frame The best example would be adjacent sides of a picture frame. Remember that adjacent means next to and sharing a vertex. The adjacent sides of a picture frame meet at a 90◦ angle and so these sides are perpendicular. Example B ← → −→ Is SO ⊥ GD? OGD ∼ = 6 SGD and the angles form a linear pair. This means both angles are 90◦ , so the lines are perpendicular. 6 Example C Write a 2-column proof to prove Theorem #1. Note: You need to understand corresponding angles in order to understand this proof. If you have not yet learned corresponding angles, be sure to check out that concept first, or skip this example for now. Given: l||m, l ⊥ n Prove: n ⊥ m 3 www.ck12.org TABLE 1.1: Statement 1. l||m, l ⊥ n 2. 6 1, 6 2, 6 3, and 6 4 are right angles 3. m6 1 = 90◦ 4. m6 1 = m6 5 5. m6 5 = 90◦ 6. m6 6 = m6 7 = 90◦ 7. m6 8 = 90◦ 8. 6 5, 6 6, 6 7, and 6 8 are right angles 9. n ⊥ m Reason 1. Given 2. Definition of perpendicular lines 3. Definition of a right angle 4. Corresponding Angles Postulate 5. Transitive PoE 6. Congruent Linear Pairs 7. Vertical Angles Theorem 8. Definition of right angle 9. Definition of perpendicular lines MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136582 CK-12 Perpendicular Lines –> Guided Practice 1. Find m6 CTA. 2. Determine the measure of 6 1. 4 www.ck12.org Chapter 1. Perpendicular Lines 3. Find m6 1. Answers: 1. These two angles form a linear pair and 6 STC is a right angle. m6 STC = 90◦ m6 CTA is 180◦ − 90◦ = 90◦ 2. We know that both parallel lines are perpendicular to the transversal. m6 1 = 90◦ . 3. The two adjacent angles add up to 90◦ , so l ⊥ m. m6 1 = 90◦ because it is a vertical angle to the pair of adjacent angles and vertical angles are congruent. Explore More Use the figure below to answer questions 1-2. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them. 5 www.ck12.org 1. List a pair of perpendicular lines. 2. For AB, how many perpendicular lines would pass through point V ? Name this/these line(s). Use the picture below for question 3. 3. If t ⊥ l, is t ⊥ m? Why or why not? Find the measure of 6 1 for each problem below. 4. 5. 6. 6 www.ck12.org Chapter 1. Perpendicular Lines 7. 8. 9. 10. 11. 12. In questions 13-16, determine if l ⊥ m. 7 www.ck12.org 13. 14. 15. 16. Fill in the blanks in the proof below. 17. Given: l ⊥ m, l ⊥ nProve: m||n 8 www.ck12.org Chapter 1. Perpendicular Lines TABLE 1.2: Statement 1. 2. 6 1 and 6 2 are right angles 3. 4. 5. m||n Reason 1. 2. 3. Definition of right angles 4. Transitive PoE 5. Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 3.2. 9