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Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY Lesson 22: Congruence Criteria for TrianglesβSAS Learning Target ο· I can explain why two triangles that satisfy the SAS congruence criterion must be congruent. Discussion Given angles β π΄ππ΅ and β π΄β² πβ² π΅ β² , how can we tell whether they have the same degree without having to measure each angle individually? Quick Write: What is congruence? ________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ http://vimeo.com/45773877 ( 2.53 mins ) http://goo.gl/Vjg9IB ( 20-30 sec) We are going to show that there are criteria that refer to a few parts (sides and/or angles) of two triangles and a correspondence between them that guarantee triangle congruency (i.e., existence of rigid motion/_____________). Side-Angle-Side triangle congruence criteria (SAS): Given two triangles β³ π¨π©πͺ and β³ π¨β²π©β²πͺβ² so that 1. π¨π© = π¨β²π©β² (Side) A B B' 2. πβ π¨ = πβ π¨β² (Angle) 3. π¨πͺ = π¨β² πͺβ² (Side). C A' Then the triangles are congruent. C' Based on the previous definition for two triangles to be congruent, we have to identify a series of rigid motion that map βπ΄β²π΅β²πΆβ² to βπ΄π΅πΆ. GOAL: To find a composition of rigid__________________ will map βπ΄β²π΅β²πΆβ² to βπ΄π΅πΆ. Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY 1. The first thing we want to do is bring to the two triangles together at a common vertex. A B B' C A' C' a. What should be the common vertex? ___________ b. What rigid motion should we use to bring those vertices together? _______________________ c. Along which vector should the rigid motion be complete? __________ β² β² β² After the ________________(below), ππ΄β²π΄ βββββββ (βπ΄ π΅ πΆ ) shares one vertex with βπ΄π΅πΆ, π΄. In fact, we can say π___________ (β_____________) = β_____________. 2. Next we want to bring two corresponding sides together. 1. What two sides should we bring together? _____________________ or ____________________ 2. What rigid motion should we use to bring those two sides together? ______________________ After a _____________________, π π΄,βπβ πΆπ΄πΆβ²β² (βπ΄π΅β²β²πΆβ²β²), a total of two vertices are shared with βπ΄π΅πΆ, π΄ and πΆ. Therefore, π ________(β³ _____________) = β³ _____________. 3. Finally, we want to map the triangles directly on top of each other. What rigid motion should we use to map the triangles on top of each other? ____________________ Since a reflection is a rigid motion and it preserves angle measures, we know that πβ π΅ β²β²β² π΄πΆ = πβ π΅π΄πΆ and βββββ . so βββββββββ π΄π΅β²β²β² maps to π΄π΅ Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY Write the transformations used to correctly notate the congruence (the composition of transformations) that takes βπ΄β² π΅ β² πΆ β² β βπ΄π΅πΆ: A 1st __________________________ B B' nd 2 __________________________ A' C 3rd __________________________ Composite Notation ______(______(______(βπ΄β² π΅ β² πΆ β² ) = βπ΄π΅πΆ C' βFollowing Notationβ ___________________ Congruence vs. Equality When writing proofs and justifying steps, itβs important to use congruent and equal appropriately. ______________ are congruent. ______________ are equal. Fill in the blanks with either = or ο AB _____ BC AB ____ BC mοABC ______mοDEF οABC ______οDEF Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria: ο· ο· ο· Use a two-column proof to state which pairs of sides or angles are congruent/equal and why If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle onto the other. When completing the proof you are looking to find a pair of sides congruent/equal, another pair of sides congruent/equal, and then the pair of included angles congruent/equal. **** In order to develop, you should remember the following: 1. Vertical angles have equal measure 2. Reflexive sides are congruent 3. Reflexive angles are congruent 4. Alternate interior angles and corresponding angles are congruent when two lines are parallel 5. Angle bisectors create congruent angles 6. Segment bisectors create congruent segments 5. Perpendicular lines create right angles NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Lesson 22 Period:________ Date:_________ M1 GEOMETRY Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria and describe the rigid motions. Example 1. Given: πβ πΏππ = πβ πΏππ, ππ = ππ. Do β³ πΏππ and β³ πΏππ meet the SAS criteria? If so, write a two column proof. If not, explain why. a) b) Describe the rigid motion: _______________________________________________________ Μ Μ Μ Μ bisect each other. Do βπ»πΊπΉ and βπΎπΊπΏ meet the SAS criteria? If so, write Example 2. Given: Μ Μ Μ Μ πΉπΏ and π»πΎ a two-column proof. If not, explain why. a) b) Describe the rigid motion: _______________________________________________________ Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ GEOMETRY Side-Angle-Side triangle congruence criteria (SAS) Given two triangles β³ π΄π΅πΆ and β³ π΄β²π΅β²πΆβ² such that: 1. π΄π΅ = π΄β²π΅β² (Side) 2. πβ π΄ = πβ π΄β² (Angle) 3. π΄πΆ = π΄β² πΆ β² (Side). Diagram: Then the triangles are congruent. ****NOTICE: _________________________________________________________________________ Use your trasperancy to help you map the β³ π΄β²π΅β²πΆβ² to β³ π΄π΅πΆ. A B B' C A' C' M1 Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY Lesson 22: Congruence Criteria for TrianglesβSAS Classwork Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria and describe the rigid motion. Μ Μ Μ bisect each other. Do βπΎπΏπ½ and βππΏπ meet the SAS criteria? If so, write a two 1. Given: Μ Μ Μ Μ Μ Μ πΎπ and Μ π½π column proof. If not, explain why. b) Describe the rigid transformation________________________________________________________ Μ Μ Μ Μ β₯ πΆπ· Μ Μ Μ Μ , π΄π΅ = πΆπ·. Do βπ΄π·π΅ and βπΆπ΅π· meet the SAS criteria? If so, write a two column 2. Given: π΄π΅ proof. If not, explain why. b) Describe the rigid transformation: ______________________________________________________ Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY Μ Μ Μ Μ , Μ Μ Μ Μ Μ Μ Μ Μ Μ . Do βπ½ππΏ and βπΎπΏπ meet the SAS criteria? If so, write a 3. Given: π½π = πΎπΏ, Μ Μ Μ Μ Μ π½π β₯ ππΏ πΎπΏ β₯ ππΏ two column proof. If not, explain why. STATEMENT REASON STATEMENT REASON 4. Given: Μ Μ Μ Μ π΅πΉ β₯ Μ Μ Μ Μ π΄πΆ , Μ Μ Μ Μ πΆπΈ β₯ Μ Μ Μ Μ π΄π΅ . Do β³ π΅πΈπ· and β³ πΆπΉπ· meet the SAS criteria? Lesson 22 NYS COMMON CORE MATHEMATICS CURRICULUM Name:___________________________________ Period:________ Date:_________ M1 GEOMETRY Lesson 22: Congruence Criteria for TrianglesβSAS Homework Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria: When completing the following proofs you are looking to find a pair of sides congruent/equal, another pair of sides congruent/equal, and then the pair of included angles congruent/equal. ****In order to find these pairs you should remember the following: 1. Vertical angles have equal measure 2. Reflexive sides are congruent 3. Reflexive angles are congruent 4. Alternate interior angles and corresponding angles are congruent when two lines are parallel 5. Angle bisectors create congruent angles 6. Segment bisectors create congruent segments 5. Perpendicular lines create right angles 1. Do βπΊπ»πΌ and βπΌπ½πΊ meet the SAS criteria? If so, write a two column proof. If not, explain why 2. Given: Μ Μ Μ Μ π΄πΈ bisects angle β π΅πΆπ·, π΅πΆ = π·πΆ. Do βπ΄π΅πΆ and βπ΄π·πΆ meet the SAS criteria? If so, write a two column proof. If not, explain why.