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Angle-Angle and Side-Side-Side Similarity Theorems CK12 Editor 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: August 11, 2012 AUTHORS CK12 Editor www.ck12.org C ONCEPT Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems 1 Angle-Angle and Side-Side-Side Similarity Theorems Lesson Plan Launch (10 min): • Students will study the ASS and AAA cases for why they don’t work • Teacher summarizes • Transition into similar triangles off of AAA example Presentation (10 min): • Go over congruency vs. similarity definitions using a t-chart to contrast • Highlight each corresponding side pair of angles or sides and then highlight that same pair in the ratio below it Practice (30 min): AA and SSS • Go over class examples and non-examples for both AA and SSS similarity. • Emphasize the stating of congruent angles in AA and the correct setup of ratios and their comparison in SSS. Conclusion (10 min): • Students will create their own examples of SSS congruence and SSS similarity. They will then use a word bank to help them explain the difference between them. Exit Ticket (5 min): Homework Materials: • Rulers • Protractors Launch Final Congruency Investigations Part 1 1 www.ck12.org TABLE 1.1: Marking S/A? Congruency Name the 3-letter congruencies group given in both triangles: ____ ____ ____ ∗ Are the two triangles congruent? Yes/No ∗ Does ASS/SSA make only 1 kind of triangle? Yes/No ∗ Therefore, if we have ASS/SSAare we sure that we have two congruent triangles? Yes/No Part 2 Draw the following 2 equiangular triangles (each angle will be 60 degrees). Label their sides and angles. TABLE 1.2: Triangle 1 Side lengths 2 cm, 2cm, 2cm Triangle 2 Side lengths 6 cm, 6 cm, 6 cm ∗ Are the triangles congruent? Yes/No ∗ Can we make only 1 kind of triangle with AAA given information? Yes/No ∗ Therefore, is AAA enough given info to know for sure that 2 triangles are congruent? Yes/No Circle the acceptable groups of evidence to say two triangles are congruent: SSS SAS ASS ASA AAS AAA Presentation Similarity (∼) Two triangles are similar if their 3 corresponding angle pairs are congruent and their 3 corresponding side pairs are proportional (same scale factor) TABLE 1.3: Congruent ∼ = Identical shape & size Angles: Congruent Sides: Congruent 2 Similar ∼ Identical shape; different sizes possible Angles: Congruent Sides: Proportional (scale factor) www.ck12.org Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems TABLE 1.3: (continued) Congruent Similar ∆STU ∼ = ∆XWV ∆STU ∼ ∆XWV Scale Factor: 3 √ 6 √3 ∆1 XW = =3 ∆2 ST 2 3 VX US = 6 2 =3 WV TU = 3 1 =3 Similarity Evidence Groups AA Similarity → If two angles in one triangle are _________ to two angles in another, then the triangles are ________. (Note: if two angle pairs are congruent, then the third angle pair must also be _______). SSS Similarity → If all 3 corresponding sides between two triangles are _________, then the triangles are ___________. Practice Directions: Are the triangles similar? Say yes or no and provide evidence. If yes, make a similarity statement like the ‘Examples’ above. 1) 3 www.ck12.org Yes/No ∆ ∼∆ by _____ 2) Yes/No ∆ ∼∆ by _____ 3) Yes/No ∆ 4) 4 ∼∆ by _____ www.ck12.org Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems ∆1 ∆2 Yes/No ∆ ∼∆ by _____ (Scale factor _____) 5) ∆1 ∆2 Yes/No ∆ ∼∆ by _____ (Scale factor _____) 6) ∆1 ∆2 5 www.ck12.org Yes/No ∆ ∼∆ by _____ (Scale factor _____) 7) Can you have two triangles that are both congruent and similar? Use the triangles at the left and your ratio tests for similarity to answer the question. Conclusion 1. Sketch and label an example of SSS Congruence. 2. Sketch and label an example of SSS Similarity. What is the difference between the two? Explain using at least 2 complete sentences and 2 of the words below: Triangles Sides Angles Congruent Similar Scale Factor Size Homework 1. Two triangles have the following side lengths: 3 cm, 5 cm, 7 cm and 6 cm, 10 cm, and 14 cm. Are the two triangles similar? Provide evidence. Similar? Say yes or no and provide evidence. If yes, make a similarity statement. 2. 6 www.ck12.org Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems 3. 4. 5. 7 www.ck12.org Exit Ticket 8