Triangle Sum Theorem. Bill Zahner Lori Jordan. Say Thanks to the Authors Click (No sign in required)

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1 Triangle Sum Theorem Bill Zahner Lori Jordan Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS Bill Zahner Lori Jordan 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, 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 (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/Share Alike 3.0 Unported (CC BY-NC-SA) License ( 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 Printed: July 22, 2012

3 1 CONCEPT 1 Triangle Sum Theorem Here you ll learn that the sum of the angles in any triangle is the same, due to the Triangle Sum Theorem. What if you wanted to classify the Bermuda Triangle by its sides and angles? You are probably familiar with the myth of this triangle; how several ships and planes passed through and mysteriously disappeared. The measurements of the sides of the triangle are in the image. What type of triangle is this? Using a protractor, find the measure of each angle in the Bermuda Triangle. What do they add up to? Do you think the three angles in this image are the same as the three angles in the actual Bermuda triangle? Why or why not? After completing this Concept, you ll be able to determine how the three angles in any triangle are related in order to help you answer these questions. Watch This MEDIA Click image to the left for more content. Now watch this video. MEDIA Click image to the left for more content. Concept 1. Triangle Sum Theorem

4 2 Guidance In polygons, interior angles are the angles inside of a closed figure with straight sides. The vertex is the point where the sides of a polygon meet. Triangles have three interior angles, three vertices and three sides. A triangle is labeled by its vertices with a. This triangle can be labeled ABC, ACB, BCA, BAC, CBA or CAB. Order does not matter. The angles in any polygon are measured in degrees. Each polygon has a different sum of degrees, depending on the number of angles in the polygon. How many degrees are in a triangle? Investigation: Triangle Tear-Up Tools Needed: paper, ruler, pencil, colored pencils 1. Draw a triangle on a piece of paper. Try to make all three angles different sizes. Color the three interior angles three different colors and label each one, 1, 2, and Tear off the three colored angles, so you have three separate angles. 3. Attempt to line up the angles so their points all match up. What happens? What measure do the three angles add up to? This investigation shows us that the sum of the angles in a triangle is 180 because the three angles fit together to form a straight line. Recall that a line is also a straight angle and all straight angles are 180. The Triangle Sum Theorem states that the interior angles of a triangle add up to 180. The above investigation is one way to show that the angles in a triangle add up to 180. However, it is not a two-column proof. Here we will prove the Triangle Sum Theorem.

5 3 Given : ABC with AD BC Prove : m 1 + m 2 + m 3 = 180 TABLE 1.1: Statement Reason 1. ABC above with AD BC Given 2. 1 = 4, 2 = 5 Alternate Interior Angles Theorem 3. m 1 = m 4,m 2 = m 5 = angles have = measures 4. m 4 + m CAD = 180 Linear Pair Postulate 5. m 3 + m 5 = m CAD Angle Addition Postulate 6. m 4 + m 3 + m 5 = 180 Substitution PoE 7. m 1 + m 3 + m 2 = 180 Substitution PoE There are two theorems that we can prove as a result of the Triangle Sum Theorem and our knowledge of triangles. Theorem #1: Each angle in an equiangular triangle measures 60. Theorem #2: The acute angles in a right triangle are always complementary. Example A What is the m T? From the Triangle Sum Theorem, we know that the three angles add up to 180. Set up an equation to solve for T. m M + m A + m T = m T = m T = 180 m T = 71 Concept 1. Triangle Sum Theorem

6 4 Example B Show why Theorem #1 is true. ABC above is an example of an equiangular triangle, where all three angles are equal. Write an equation. m A + m B + m C = 180 m A + m A + m A = 180 3m A = 180 m A = 60 If m A = 60, then m B = 60 and m C = 60. Example C Use the picture below to show why Theorem #2 is true. m O = 41 and m G = 90 because it is a right angle. m D + m O + m G = 180 m D = 180 m D + 41 = 90 m D = 49 Notice that m D + m O = 90 because G is a right angle. Watch this video for help with the Examples above.

7 5 Concept Problem Revisited The Bermuda Triangle is an acute scalene triangle. The angle measures are in the picture to the right. Your measured angles should be within a degree or two of these measures. The angles should add up to 180. However, because your measures are estimates using a protractor, they might not exactly add up. The angle measures in the picture are the actual measures, based off of the distances given, however, your measured angles might be off because the drawing is not to scale. Vocabulary A triangle is a three sided shape. All triangles have three interior angles, which are the inside angles connecting the sides of the triangle. The vertex is the point where the sides of a polygon meet. Special types of triangles are listed below: Scalene: All three sides are different lengths. Isosceles: At least two sides are congruent. Equilateral: All three sides are congruent. Right: One right angle. Acute: All three angles are less than 90. Obtuse: One angle is greater than 90. Equiangular: All three angles are congruent. Guided Practice 1. Determine m 1 in this triangle: 2. Two interior angles of a triangle measure 50 and 70. What is the third interior angle of the triangle? Concept 1. Triangle Sum Theorem

8 Find the value of x and the measure of each angle. Answers: m 1 = 180. Solve this equation and you find that m 1 = x = 180. Solve this equation and you find that the third angle is All the angles add up to 180. (8x 1) + (3x + 9) + (3x + 4) = 180 (14x + 12) = x = 168 x = 12 Substitute in 12 for x to find each angle. [3(12) + 9] = 45 [3(12) + 4] = 40 [8(12) 1] = 95 Practice Determine m 1 in each triangle

9 Two interior angles of a triangle measure 32 and 64. What is the third interior angle of the triangle? 9. Two interior angles of a triangle measure 111 and 12. What is the third interior angle of the triangle? 10. Two interior angles of a triangle measure 2 and 157. What is the third interior angle of the triangle? Find the value of x and the measure of each angle Concept 1. Triangle Sum Theorem

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