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Angle Measures in Given Quadrilaterals Jen Kershaw Kimberly Hopkins 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 © 2016 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: July 10, 2016 AUTHORS Jen Kershaw Kimberly Hopkins www.ck12.org C HAPTER Chapter 1. Angle Measures in Given Quadrilaterals 1 Angle Measures in Given Quadrilaterals In this concept, you will learn how to work with the angle measures of a quadrilateral. . Ivonne is a farmer. She plants a rhombus-shaped section of corn each year. This year, she has decided to check the angles at each vertex of the corn. She determines that one angle is 110 degrees and the another angle is 70 degrees. Her daughter tells her that she can determine the other angle measures without walking around to them. What are the other angle measures? In this concept, you will learn how to work with the angle measures of a quadrilateral. Measuring Angles in Quadrilaterals Let’s look at a square. 1 www.ck12.org A square has four right angles. Each right angle is 90◦ . You can add up the sum of the interior angles of a square and see how this is related to all quadrilaterals. 90 + 90 + 90 + 90 = 360◦ The sum of the interior angles of all quadrilaterals is 360◦ . You can use this information to find the measure of missing angles. Write an equation using the variable and given measurements and figure out the measure of the missing angle. 80 + 75 + 105 + x = 360 260 + x = 360 360 − 260 = x 100 = x The missing angle is equal to 100◦ . Use this information to help you when figuring out missing angle measures in different quadrilaterals. Also, you can draw specific quadrilaterals using a ruler and a protractor. Use the protractor to be sure that your work is accurate. This is especially important when drawing squares or rectangles or any figure with a right angle. Start by using a protractor to draw in each of the four right angles. By using a ruler and a protractor, your lines will be straight and we will be able to determine that we have drawn the square correctly. Drawing it freehand may seem easier, but it does not assure accuracy! The best way to be sure that your work is accurate is to use a protractor and a ruler. 2 www.ck12.org Chapter 1. Angle Measures in Given Quadrilaterals Here is the first angle of a square. Now turn the protractor upside down and draw the other angle. Here is the final figure. Examples Example 1 Earlier, you were given a problem about Ivonne and her rhombus-shaped field of corn. 3 www.ck12.org Ivonne knows that one of the angles of the field is 110 degrees and a different angle is 80 degrees. What are the other angle measures? First, note the properties of angles in a rhombus. Opposite sides are congruent Next, list all of the angle values. 80, 80, 110, and 110 Then, state the other angle measures. 80 and 110 The answer is 80 degrees and 110 degrees. Example 2 A quadrilateral has the following angle measures: 130, 80 and 95. What is the measure of the missing angle? First, write an equation. 130 + 80 + 95 + x = 360 Then, solve for the missing angle. 55 The answer is 55 degrees. Example 3 If one angle of a rectangle is 90 degrees, what are the measures of the other three angles? First, remember the angle rules for a rectangle. All of the angles are 90 degrees Then, state the measures of the other angles, 90 degrees The answer is 90 degrees. Example 4 A quadrilateral has the following angle measures: 105, 90 and 88. What is the measure of the missing angle? First, write an equation. 105 + 90 + 88 + x = 360 Then, solve for the missing angle. 77 The answer is 77 degrees. 4 www.ck12.org Chapter 1. Angle Measures in Given Quadrilaterals Example 5 A parallelogram has two congruent angles that are both 85 degrees. The other two angles are congruent. What is the measure of each missing angle? First, write an equation. 85 + 85 + x + x = 360 Next, simplify the equation. 170 + 2x = 360 Then, solve for the missing angle. 2x = 190 x = 95 The answer is 95 degrees. Review Answer each of the following questions about quadrilaterals. 1. 2. 3. 4. 5. 6. True or false. A quadrilateral will always have only four sides. The interior angles of a quadrilateral add up to be _________ degrees. A square will have four ___________ degree angles. A rectangle will have four ___________ degree angles. True or false. A rhombus will also always have four right angles. If the sum of three of the angles of a quadrilateral is equal to 300◦ , it means that the measure of the missing angle is ____________. 7. What is the value of x? 8. 5 www.ck12.org 9. 10. What are all four angles of this rectangle equal to? 11. If the sum of the interior angles of a quadrilateral is equal to 360◦ , how many triangles can you draw inside a quadrilateral? 12. How many degrees are in a triangle? 13. Write an equation to show how the angles of the two triangles are equal to 360 degrees. Identify the following figures. 14. 15. Review (Answers) To see the Review answers, open this PDF file and look for section 9.11. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/181940 6 www.ck12.org Chapter 1. Angle Measures in Given Quadrilaterals 7