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Isosceles Triangles Andrew Gloag Bill Zahner Dan Greenberg Jim Sconyers Lori Jordan Victor Cifarelli 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: October 30, 2012 AUTHORS Andrew Gloag Bill Zahner Dan Greenberg Jim Sconyers Lori Jordan Victor Cifarelli EDITOR Annamaria Farbizio www.ck12.org C ONCEPT Concept 1. Isosceles Triangles 1 Isosceles Triangles Here you’ll learn the definition of an isosceles triangle as well as two theorems about isosceles triangles: 1) The angle bisector of the vertex is the perpendicular bisector of the base; and 2) The base angles are congruent. What if you were presented with an isoceles triangle and told that its base angles measure x◦ and y◦ ? What could you conclude about x and y? After completing this Concept, you’ll be able to apply important properties about isoceles triangles to help you solve problems like this one. Watch This MEDIA Click image to the left for more content. CK-12 Watch the first part of this video. MEDIA Click image to the left for more content. James Sousa:HowTo Construct AnIsosceles Triangle Then watch this video. MEDIA Click image to the left for more content. James Sousa:Proof of the Isosceles TriangleTheorem Finally, watch this video. MEDIA Click image to the left for more content. James Sousa:Using the Properties ofIsosceles Trianglesto Determine Values 1 www.ck12.org Guidance An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle are called the legs. The other side is called the base. The angles between the base and the legs are called base angles. The angle made by the two legs is called the vertex angle. One of the important properties of isosceles triangles is that their base angles are always congruent. This is called the Base Angles Theorem. For 4DEF, if DE ∼ = EF, then 6 D ∼ = 6 F. Another important property of isosceles triangles is that the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is called the Isosceles Triangle Theorem. (Note this is ONLY true of the vertex angle.) The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true as well. Base Angles Theorem Converse: If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. So for 4DEF, if 6 D ∼ = 6 F, then DE ∼ = EF. Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle. So for isosceles 4DEF, if EG ⊥ DF and DG ∼ = GF, then 6 DEG ∼ = 6 FEG. Example A Which two angles are congruent? This is an isosceles triangle. The congruent angles are opposite the congruent sides. From the arrows we see that 6 S∼ = 6 U. Example B If an isosceles triangle has base angles with measures of 47◦ , what is the measure of the vertex angle? Draw a picture and set up an equation to solve for the vertex angle, v. Remember that the three angles in a triangle always add up to 180◦ . 47◦ + 47◦ + v = 180◦ v = 180◦ − 47◦ − 47◦ v = 86◦ Example C If an isosceles triangle has a vertex angle with a measure of 116◦ , what is the measure of each base angle? Draw a picture and set up and equation to solve for the base angles, b. 116◦ + b + b = 180◦ 2b = 64◦ b = 32◦ 2 www.ck12.org Concept 1. Isosceles Triangles MEDIA Click image to the left for more content. CK-12 Isosceles Triangles Guided Practice 1. Find the value of xand the measure of each angle. 2. Find the measure of x. 3. True or false: Base angles of an isosceles triangle can be right angles. Answers: 1. The two angles are equal, so set them equal to each other and solve for x. (4x + 12)◦ = (5x − 3)◦ 15 = x Substitute x = 15; the base angles are [4(15) + 12]◦ , or 72◦ . The vertex angle is 180◦ − 72◦ − 72◦ = 36◦ . 2. The two sides are equal, so set them equal to each other and solve for x. 2x − 9 = x + 5 x = 14 3. This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right (90◦ ) angles. The Triangle Sum Theorem states that the sum of the three angles in a triangle is 180◦ . If two of the angles in a triangle are right angles, then the third angle must be 0◦ and the shape is no longer a triangle. Practice Find the measures of x and/or y. 1. 2. 3. 4. 5. Determine if the following statements are true or false. 6. 7. 8. 9. Base angles of an isosceles triangle are congruent. Base angles of an isosceles triangle are complementary. Base angles of an isosceles triangle can be equal to the vertex angle. Base angles of an isosceles triangle are acute. 3 www.ck12.org Fill in the proofs below. 10. Given: Isosceles 4CIS, with base angles 6 C and 6 SIO is the angle bisector of 6 CISProve: IO is the perpendicular bisector of CS TABLE 1.1: Statement 1. 2. 3. 6 CIO ∼ = 6 SIO 4. 5. 4CIO ∼ = 4SIO ∼ 6. CO = OS 7. 8. 6 IOC and 6 IOS are supplementary 9. 10. IO is the perpendicular bisector of CS Reason 1. Given 2. Base Angles Theorem 3. 4. Reflexive PoC 5. 6. 7. CPCTC 8. 9. Congruent Supplements Theorem 10. 11. Given: Isosceles 4ICS with 6 C and 6 SIO is the perpendicular bisector of CSProve: IO is the angle bisector of 6 CIS TABLE 1.2: Statement 1. 2. 6 C ∼ =6 S 3. CO ∼ = OS 6 4. m IOC = m6 IOS = 90◦ 5. 6. 7. IO is the angle bisector of 6 CIS Reason 1. 2. 3. 4. 5. 6. CPCTC 7. On the x − y plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle. 12. 13. 14. 15. 16. 4 (-2, 1), (1, -2), (-5, -2) (-2, 5), (2, 4), (0, -1) (6, 9), (12, 3), (3, -6) (-10, -5), (-8, 5), (2, 3) (-1, 2), (7, 2), (3, 9)