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Elizabeth Pawelka Triangle Congruence (SSS and SAS) 2/24/12 p.1 Geometry Lesson Plans Section 4-2: Triangle Congruence by SSS and SAS 2/24/12 Statement of Objectives The student will be able to… prove triangles congruent using the SSS and SAS Postulates Warm-up and Review of Homework (25 minutes) Warm-up, practice book p. 40, 1-10 (answers on sheet) (10 minutes) They do warm-up while I check homework from previous night to see if there are any common errors I should address during review Go over Warm-up (5 minutes) Homework from last night: (pp. 182 – 184: 3-28, 30-35, 38-40, 44) ask for any specific questions and specifically go over #28 and # 44 (10 minutes) #28: a. Given a. AB || DC b. Alternate Interior Angles Thm b. ∠CAB ≅∠ACD c. ∠B ≅∠D c. Given d. ∠BCA ≅∠DAC d. If 2 ∠’s ≅ then 3rd∠’s ≅ Thm e. AC ≅ CA e. Reflexive Property f. AB ≅ CD , BC ≅ AD f. Given g. ΔABC ≅ ΔCDA g. Definition of ≅ Δs #44 a. PR || TQ a. Given b. ∠PRS ≅∠QTS b. Alternate Interior Angles Thm c. ∠RPS ≅∠TQS c. Alternate Interior Angles Thm d. ∠PSR ≅∠QST d. Vertical ∠’s ≅ e. Given e. PR ≅ QT , PS ≅ QS f. PQ bisects RT f. Given g. RS ≅ TS g. Def of segment bisector h. ΔPRS ≅ ΔQTS h. Definition of ≅ Δs Elizabeth Pawelka Triangle Congruence (SSS and SAS) 2/24/12 p.2 Teacher Input (60 minutes) Activity (20 minutes) Introduce SSS and SAS with an investigation using the Illuminations Congruence Applet: http://illuminations.nctm.org/ActivityDetail.aspx?ID=4 – Do examples with 3 sides (SSS), with 2 sides and included angle (SAS), and with 2 sides and excluded angle (SSA) to show that they aren’t always congruent. Do the first one and then have different students come up to make congruent triangles and to try to make non-congruent triangles. Lecture/Examples: (40 minutes) In which configuration(s) were the triangles always congruent? i. Side-Angle-Side (SAS) ii. Side-Side-Side (SSS) In which configuration(s) sometimes congruent? i. Side-Side-Angle (SSA) Since proofs have been problematic, include different forms of proofs (2-column/flow/paragraph) and point out that there are only 5 types of reasons: Given, Definitions, Properties, Postulates, and Theorems. Introduce the SSS Theorem: “If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.” Include a diagram to illustrate. Example 1: Prove: ΔABD ≅ ΔCBD Given: ≅ segments as marked Proof: AB ≅ CB Given DB ≅ DB Reflexive AD ≅ CD Given ΔABD ≅ ΔCBD by SSS Example 2: Prove ΔADE ≅ ΔCBE Given: Congruent sides as marked and AC and BD bisect each other Write on board that AE = EC and BE = ED and so Elizabeth Pawelka Triangle Congruence (SSS and SAS) 2/24/12 p.3 Example 3: To Prove ΔADC ≅ ΔBDC What else do we know? What else do we need to know? by Reflexive Property to prove by SSS Introduce SAS Postulate: “If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.” Include a diagram to illustrate. Example 4: Prove ΔABD ≅ ΔCBD – I do one 1. AB ≅ BC ; ∠ABD ≅∠DBC 1. Given 2. DB ≅ DB 2. Reflexive Property 3. ΔABD ≅ ΔCBD 3. SAS Postulate Example 5: Prove ΔDFE ≅ ΔHFG – They do one 1. EF ≅ FG ; DF ≅ FH 1. Given 2. ∠DFE ≅ ∠HFG 2. Vertical ∠’s are ≅ 3. ΔDFE ≅ ΔHFG 3. SAS Postulate Example 6: Prove: ΔAEB ≅ ΔCED Given: The diagonal legs are joined at their midpoints Since the legs are joined at their midpoints, AE ≅ EC and BE ≅ DE . ∠AEB ≅ ∠CED because they are vertical ∠’s. Therefore ΔAEB ≅ ΔCED by SAS. ■ Elizabeth Pawelka Triangle Congruence (SSS and SAS) 2/24/12 p.4 Example 7: What is the congruency statement? ΔCAT ≅ ΔDOG Closure (5 minutes) Today we learned to recognize congruent figures and corresponding parts Tomorrow we’ll learn how to prove triangles congruent using the SSS and SAS Postulates Homework (attached) pp. 189 – 192: 1-30, 33, 41-44