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Vertical Angles Bill Zahner Lori Jordan 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-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2012 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/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/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/terms. Printed: December 19, 2012 AUTHORS Bill Zahner Lori Jordan www.ck12.org C ONCEPT Concept 1. Vertical Angles 1 Vertical Angles Here you’ll learn about vertical angles and how they can help you to solve problems in geometry. What if you want to know how opposite pairs of angles are related when two lines cross, forming four angles? After completing this Concept, you’ll be able to apply the properties of these special angles to help you solve problems in geometry. Watch This MEDIA Click image to the left for more content. CK-12 Foundation: Chapter1VerticalAnglesA MEDIA Click image to the left for more content. James Sousa:Vertical Angles Guidance Vertical angles are two non-adjacent angles formed by intersecting lines. In the picture below, 6 1 and 6 3 are vertical angles and 6 2 and 6 4 are vertical angles. Notice that these angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol. 1 www.ck12.org Investigation: Vertical Angle Relationships 1. Draw two intersecting lines on your paper. Label the four angles created 6 1, 6 2, 6 3, and 6 4. See the picture above. 2. Take your protractor and find m6 1. 3. What is the angle relationship between 6 1 and 6 2? Find m6 2. 4. What is the angle relationship between 6 1 and 6 4? Find m6 4. 5. What is the angle relationship between 6 2 and 6 3? Find m6 3. 6. Are any angles congruent? If so, write down the congruence statement. From this investigation, hopefully you found out that 6 1 ∼ = 6 3 and 6 2 ∼ = 6 4. This is our first theorem. That means it must be proven true in order to use it. Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. We can prove the Vertical Angles Theorem using the same process we used above. However, let’s not use any specific values for the angles. From the picture above: 1 and 6 2 are a linear pair m6 1 + m6 2 = 180◦ 6 2 and 6 3 are a linear pair m6 2 + m6 3 = 180◦ 6 3 and 6 4 are a linear pair m6 3 + m6 4 = 180◦ 6 All of the equations = 180◦ , so set the m6 1 + m6 2 = m6 2 + m6 3 first and second equation equal to AND each other and the second and third. m6 2 + m6 3 = m6 3 + m6 4 Cancel out the like terms m6 1 = m6 3, m6 2 = m6 4 Recall that anytime the measures of two angles are equal, the angles are also congruent. Example A Find m6 1 and m6 2. 1 is vertical angles with 18◦ , so m6 1 = 18◦ . 180◦ − 18◦ = 162◦ . 6 6 2 is a linear pair with 6 1 or 18◦ , so 18◦ + m6 2 = 180◦ . m6 2 = Example B Name one pair of vertical angles in the diagram below. 2 www.ck12.org Concept 1. Vertical Angles One example is 6 INJ and 6 MNL. Example C If 6 ABC and 6 DBF are vertical angles and m6 ABC = (4x + 10)◦ and m6 DBF = (5x + 2)◦ , what is the measure of each angle? Vertical angles are congruent, so set the angles equal to each other and solve for x. Then go back to find the measure of each angle. 4x + 10 = 5x + 2 x=8 So, m6 ABC = m6 DBF = (4(8) + 10)◦ = 42◦ Watch this video for help with the Examples above. MEDIA Click image to the left for more content. CK-12 Foundation: Chapter1VerticalAnglesB Vocabulary Vertical angles are two non-adjacent angles formed by intersecting lines. They are always congruent. Guided Practice Find the value of x or y. 1. 2. 3 www.ck12.org 3. Answers: 1. Vertical angles are congruent, so set the angles equal to each other and solve for x. x + 16 = 4x − 5 3x = 21 x = 7◦ 2. Vertical angles are congruent, so set the angles equal to each other and solve for y. 9y + 7 = 2y + 98 7y = 91 y = 13◦ 3. Vertical angles are congruent, so set the angles equal to each other and solve for y. 11y − 36 = 63 11y = 99 y = 9◦ Practice ← → Use the diagram below for exercises 1-2. Note that NK ⊥ IL . 1. Name one pair of vertical angles. 2. If m6 INJ = 53◦ , find m6 MNL. 4 www.ck12.org Concept 1. Vertical Angles For exercises 3-5, determine if the statement is true or false. 3. Vertical angles have the same vertex. 4. Vertical angles are supplementary. 5. Intersecting lines form two pairs of vertical angles. Solve for the variables. 6. Find x. 7. Find y. 8. 9. 10. 11. 12. If 6 ABC and 6 DBF are vertical angles and m6 ABC = (4x + 1)◦ and m6 DBF = (3x + 29)◦ , what is the measure of each angle? 13. If 6 ABC and 6 DBF are vertical angles and m6 ABC = (5x + 2)◦ and m6 DBF = (6x − 32)◦ , what is the measure of each angle? 5 www.ck12.org 14. If 6 ABC and 6 DBF are vertical angles and m6 ABC = (x + 20)◦ and m6 DBF = (4x + 2)◦ , what is the measure of each angle? 15. If 6 ABC and 6 DBF are vertical angles and m6 ABC = (2x + 10)◦ and m6 DBF = (4x)◦ , what is the measure of each angle? 6