Lesson 24: Congruence Criteria For Triangles—ASA And SSS
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWLesson 24: Congruence Criteria for Triangles—ASA and SSSStudent Outcomes Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be congruent.Lesson NotesThis is the third lesson in the Congruency topic. So far students have studied the SAS triangle congruence criteria andhow to prove base angles of an isosceles triangle congruent. Students examine two more triangle congruence criteria inthis lesson: ASA and SSS. Each proof assumes the initial steps from the proof of SAS; ask students to refer to their noteson SAS to recall these steps before proceeding with the rest of the proof. Exercises will require the use of all threetriangle congruence criteria.ClassworkOpening Exercise (7 minutes)hƐĞ ƚŚĞ ƉƌŽǀŝĚĞĚ ϯϬȗ ĂŶŐůĞ ĂƐ ŽŶĞ ďĂƐĞ ĂŶŐůĞ ŽĨ ĂŶ ŝƐŽƐĐĞůĞƐ ƚƌŝĂŶŐůĞ͘ hƐĞ Ă ĐŽŵƉĂƐƐ ĂŶĚ ƐƚƌĂŝŐŚƚ ĞĚŐĞ ƚŽ ĐŽŶƐƚƌƵĐƚ ĂŶ appropriate isosceles triangle around it.Compare your constructed isoƐĐĞůĞƐ ƚƌŝĂŶŐůĞ ǁŝƚŚ Ă ŶĞŝŐŚďŽƌ͛Ɛ͘ ŽĞƐ ƚŚĞ ƵƐĞ ŽĨ Ă ŐŝǀĞŶ ĂŶŐůĞ ŵĞĂƐƵƌĞ ŐƵĂƌĂŶƚĞĞ ƚŚĂƚ Ăůů the triangles constructed in class have corresponding sides of equal lengths?No; side lengths may vary.Discussion (25 minutes)Today we are going to examine two more triangle congruence criteria, Angle-Side-Angle (ASA) and Side-Side-Side (SSS), toĂĚĚ ƚŽ ƚŚĞ ĐƌŝƚĞƌŝĂ ǁĞ ŚĂǀĞ ĂůƌĞĂĚLJ ůĞĂƌŶĞĚ͘ tĞ ďĞŐŝŶ ǁŝƚŚ ƚŚĞ ĐƌŝƚĞƌŝĂ͘Angle-Side-Angle triangle congruence criteria (ASA): Given two triangles A ĂŶĚ ͛ ͛ ͛. If ס ס ᇱ ᇱ ᇱ (Angle), Ԣ Ԣ (Side), and ס ס ᇱ ᇱ ᇱ (Angle), then the triangles are congruent.Lesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.189
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWProoftĞ ĚŽ ŶŽƚ ďĞŐŝŶ Ăƚ ƚŚĞ ǀĞƌLJ ďĞŐŝŶŶŝŶŐ ŽĨ ƚŚŝƐ ƉƌŽŽĨ͘ ZĞǀŝƐŝƚ LJŽƵƌ ŶŽƚĞƐ ŽŶ ƚŚĞ ƉƌŽŽĨ ĂŶĚ ƌĞĐĂůů ƚŚĂƚ ƚŚĞƌĞ ĂƌĞ ƚŚƌĞĞ cases to consider when comparing two triangles. In the most general of cases, when comparing two distinct triangles, wetranslate one ǀĞƌƚĞdž ƚŽ ĂŶŽƚŚĞƌ ;ĐŚŽŽƐĞ ĐŽŶŐƌƵĞŶƚ ĐŽƌƌĞƐƉŽŶĚŝŶŐ ĂŶŐůĞƐͿ͘ ƌŽƚĂƚŝŽŶ ďƌŝŶŐƐ ĐŽŶŐƌƵĞŶƚ ĐŽƌƌĞƐƉŽŶĚŝŶŐ ƐŝĚĞƐ ƚŽŐĞƚŚĞƌ͘ ŝŶĐĞ ƚŚĞ ĐƌŝƚĞƌŝĂ ĂůůŽǁƐ ĨŽƌ ƚŚĞƐĞ ƐƚĞƉƐ͕ ǁĞ ďĞŐŝŶ ŚĞƌĞ͘In order to map ᇞ ԢԢԢ to ᇞ , we apply a reflection across the line . A reflection will map to and to ,since they are on line . However, we will say that ( ԢԢԢ) ;כ though we know that ( ԢԢԢ) is now in the same halfplane of line as , we cannot assume that ᇱᇱᇱ maps to . So we have (ᇞ ԢԢԢ) ᇞ כ . To prove the theorem,we need to verify that כ is .By hypothesis, we know that ס ס ԢԢԢ (recall that ס ԢԢԢ is the result of two rigid motions of ס ᇱ ᇱ ᇱ , so musthave the same angle measure as ס ᇱ ᇱ ᇱ ). Similarly, ס ס ԢԢԢ . Since ס ( ס ᇱᇱᇱ ) כ ס , and and ሬሬሬሬሬሬሬറ ͕ כ ŵƵƐƚ ĂĐƚƵĂůůLJ ďĞ ƚŚĞ ƐĂŵĞ ƌĂLJ͘ and כ are in the same half-plane of line , we conclude that the rays, ሬሬሬሬሬറሬሬሬሬሬറ, the point כ ŵƵƐƚ ďĞ Ă ƉŽŝŶƚ ŽŶ ƚŚĞ ƌĂLJ ሬሬሬሬሬറ somewhere. UsingBecause the points and כ define the same ray as ሬሬሬሬሬሬറ and ሬሬሬሬሬሬሬറ ͕ כ ŵƵƐƚ ďĞ the second equality of angles, ס ( ס ᇱᇱᇱ ) כ ס , we can also conclude that the rays, כ כ �ሬሬറthe same ray. Therefore, the point ŵƵƐƚ ĂůƐŽ ďĞ ŽŶ ƚŚĞ ƌĂLJ . Since ŝƐ ŝŶ ďŽƚŚ ŽĨ ƚŚĞ ƌĂLJƐ͕ and , and thetwo rays only have one point in common, namely , we conclude that כ .We have now used a series of rigid motions to map two triangles that meet the ASA criteria onto one another.Side-Side-Side triangle congruence criteria (SSS): 'ŝǀĞŶ ƚǁŽ ƚƌŝĂŶŐůĞƐ ĂŶĚ ͛ ͛ ͛͘ /Ĩ ۯ ۰ ۯ Ԣ۰Ԣ (Side), ۯ ۯ Ԣ Ԣ(Side), and ۰ ۰Ԣ Ԣ (Side) then the triangles are congruent.Lesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.190
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWProof ŐĂŝŶ͕ ǁĞ ĚŽ ŶŽƚ ƐƚĂƌƚ Ăƚ ƚŚĞ ďĞŐŝŶŶŝŶŐ ŽĨ ƚŚŝƐ ƉƌŽŽĨ͕ ďƵƚ ĂƐƐƵŵĞ ƚŚĞƌĞ ŝƐ Ă ĐŽŶŐƌƵĞŶĐĞ ƚŚĂƚ ďƌŝŶŐƐ Ă ƉĂŝƌ ŽĨ ĐŽƌƌĞƐƉŽŶĚŝŶŐ sides together, namely the longest side of each triangle.tŝƚŚŽƵƚ ĂŶLJ ŝŶĨŽƌŵĂƚŝŽŶ ĂďŽƵƚ ƚŚĞ ĂŶŐůĞƐ ŽĨ ƚŚĞ ƚƌŝĂŶŐůĞƐ͕ ǁĞ ĐĂŶŶŽƚ ƉĞƌĨŽƌŵ Ă ƌĞĨůĞĐƚŝŽŶ ĂƐ ǁĞ ŚĂǀĞ ŝŶ ƚŚĞ ƉƌŽŽĨƐ ĨŽƌ SAS and ASA. What can we do? First we add a construction: draw an auxiliary line from to ͕͛ ůĂďĞůŝŶŐ ƚŚĞ ĂŶŐůĞƐ ĐƌĞĂƚĞĚ ďLJ ƚŚĞ ĂƵdžiliary line as , , , and .Since ۯ ۰ Ԣ and ۰ Ԣ, ᇞ Ԣ and ᇞ Ԣ ĂƌĞ ďŽƚŚ ŝƐŽƐĐĞůĞƐ ƚƌŝĂŶŐůĞƐ ƌĞƐƉĞĐƚŝǀĞůLJ ďLJ ĚĞĨŝŶŝƚŝŽŶ͘ Therefore, , ďĞĐĂƵƐĞ ƚŚĞLJ ĂƌĞ ďĂƐĞ ĂŶŐůĞƐ ŽĨ an isosceles triangle ᇞ Ԣ. Similarly, , ďĞĐĂƵƐĞ ƚŚĞLJ ĂƌĞ ďĂƐĞ angles ofᇞ Ԣ. Hence, ס ס ᇱ . Since ס ס Ԣ , we say that ᇞ ᇞ Ԣ ďLJ ͘We have now used a series of rigid motions and a construction to map two triangles that meet the SSS criteria onto oneanother.Now we have three triangle congruence criteria at our disposal: SAS, ASA, and SSS. We will use these criteria todetermine whether or not pairs of triangles are congruent.Lesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.191
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWExercises (6 minutes)Based on the information provided, deƚĞƌŵŝŶĞ ǁŚĞƚŚĞƌ Ă ĐŽŶŐƌƵĞŶĐĞ ĞdžŝƐƚƐ ďĞƚǁĞĞŶ ƚƌŝĂŶŐůĞƐ͘ /Ĩ Ă ĐŽŶŐƌƵĞŶĐĞ ďĞƚǁĞĞŶ triangles exists, or if multiple congruencies exist, state the congruencies and the criteria used to determine them.1.Given: is the midpoint of , ס ס .ᇞ ᇞ , ASA2.Given:Rectangle with diagonal .ᇞ ᇞ , SSS/SAS/ASA3.Given: , .ᇞ ᇞ , SASᇞ ᇞ , SAS4.Given: ס ס , .ᇞ ᇞ , SASᇞ ᇞ , SAS/ASA5.Given: , , .ᇞ ᇞ , SAS Exit TicketLesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.192
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWNameDateLesson 24: Congruence Criteria for Triangles—ASA and SSSExit TicketBased on the information provided, determine whether a congruence exists between triangles. If a congruence betweentriangles exists, or if multiple congruencies exist, state the congruencies and the criteria used to determine them.Given: ܦܥ ܦܤ , ܧ is the midpoint of ܥܤ .Lesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.193
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEWExit Ticket Sample SolutionsThe following solutions indicate an understanding of the objectives of this lesson: ĂƐĞĚ ŽŶ ƚŚĞ ŝŶĨŽƌŵĂƚŝŽŶ ƉƌŽǀŝĚĞĚ͕ ĚĞƚĞƌŵŝŶĞ ǁŚĞƚŚĞƌ Ă ĐŽŶŐƌƵĞŶĐĞ ĞdžŝƐƚƐ ďĞƚǁĞĞŶ ƚƌŝĂŶŐůĞƐ͘ /Ĩ Ă ĐŽŶŐƌƵĞŶĐĞ ďĞƚǁĞĞŶ triangles exists, or if multiple congruencies exist, state the congruencies and the criteria used to determine them.Given: , is the midpoint of .ᇞ ᇞ , SASWƌŽďůĞŵ Ğƚ ĂŵƉůĞ SolutionsUse your knowledge of ƚƌŝĂŶŐůĞ ĐŽŶŐƌƵĞŶĐĞ ĐƌŝƚĞƌŝĂ ƚŽ ǁƌŝƚĞ ƉƌŽŽĨƐ ĨŽƌ ĞĂĐŚ ŽĨ ƚŚĞ ĨŽůůŽǁŝŶŐ ƉƌŽďůĞŵƐ͘1.Given:Prove:CA DACircles with centers and intersect at and . ס ס .Radius of CircleCB DBRadius of CircleAB ABCommon Sideο ο SSS ס ס Corr. ס ο2.Given:Prove: ס ס ס ס , , . .Given Given JL JK KLSegments AddKM KL LMSegments AddJL KMSubstitutionο ο SASKR LRBase ס converse ס ס GivenCorr. ס οLesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.194
Lesson 24NYS COMMON CORE MATHEMATICS CURRICULUMM1GEOMETRYPENDING FINAL EDITORIAL REVIEW3. ס ס and ࢠס ס .Given:(1) ᇞ ᇞ Prove:(2) and ٣ ࢠס ס Given ס ס Supplements of ס ס ס Given ο ο ASA corr. οCommon Side ס ס corr. ס ο ο ο ס ס ૡ ǏCommon SideSAS ס on a line ( ) ס ૡ ૢ ס Ǐ ٣ Def. of ٣Lesson 24:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgCongruence Criteria for Triangles—ASA and SSS7/10/13This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.195
Lesson 24: Congruence Criteria for Triangles—ASA and SSS Student Outcomes Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be congruent. Lesson Notes This is the third lesson in the Congruency topic. So far students have studied the SAS triangle congruence criteria and
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
Right Triangle Congruence Theorems Vocabulary Choose the diagram that models each right triangle congruence theorem. 1. Hypotenuse-Leg (HL) Congruence Theorem a. X Y Z Q R P 2. Leg-Leg (LL) Congruence Theorem b. U V X W 3. Hypotenuse-Angle
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Archaeological Research & Consultancy at the University of Sheffield Research School of Archaeology West Court 2 Mappin Street Sheffield S1 4DT Phone 0114 2225106 Fax 0114 2797158 Project Report 413h.1 Archaeological Evaluation of the Upper Loading Bay, Castle Market, Sheffield April 2002 By Glyn Davies and James Symonds With Contributions by Chris Cumberpatch, Jennie Stopford, Hugh Willmott .
