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LP-W10-Q2-01/17/2012 What’s man’s first duty? The answer’s brief: to be himself Henrik Ibsen, 1828-1906 Norwegian writer, dramatist, poet. _________________________________________________ GEOMETRY 01/17/2012 www.miamiseniorhs.com LESSON PLAN RESOURCES.xlsx2 MA.912.A.3.1 MA.912.A.3.9 MA.912.A.3.12 MA.912.G.1.1 MA.912.G.1.3 MA.912.G.1.4 MA.912.G.4.1 MA.912.G.4.2 MA.912.G.8.4 MA.912.G.8.6 Chapter 4: Proving Triangle Congruency Sections: 4.4 Are there Congruence Shortcuts? 4.5 Are There Other Congruence Shortcuts? Objectives: -Apply postulates and theorems to prove triangle congruency. SAS, ASA, AAS, SSS 1 ASSIGNMENTS: 4.4 and 4.5 1. Proving Congruency. Examples. 2. Interactive practice. (KA) 3. DG Workbook Pgs. 26, 27 and 28 4. Homework: DG Pgs. 24 and 25 SSS Postulate. If three sides of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. 1. Discussion of conjectures, definitions and examples. 1. Given: X is the midpoint of ̅̅̅̅ 𝑨𝑩 and ̅̅̅̅ 𝑪𝑫. AC = BD Prove: AXC ≅ BXD A (IA) C X D B 2 STATEMENTS 1. CX = DX 2. AX = BX 3. AC = BD 4. AXC ≅ BXD REASONS GIVEN GIVEN GIVEN SSS POSTULATE ̅̅̅̅ 2. Given: S is the midpoint of 𝑻𝑽 TR = VR Prove: TSR ≅ VSR T STATEMENTS 1. TS = VS 2. RS = RS 3. TR = VR 4. TSR ≅ VSR (IA) R S V REASONS GIVEN REFLEXIVE PROPERTY GIVEN SSS POSTULATE 3 NOW TRY # 3g 3. Given: ZY = WX ZW = YZ Prove: TRS ≅ Z Y VSR W STATEMENTS 1. ZY = WX; ZW = YX 2. WY = WY 3. WZY ≅ YXW X REASONS GIVEN REFLEXIVE PROPERTY SSS POSTULATE 4. Suppose you wish to use the SSS Postulate to prove that DEF ≅ JKM. What value must x have? (IA) Reasoning: To prove triangle congruency using SSS Postulate, we must be aware of that all corresponding sides of the triangles are congruent. Thus, FD = MJ, DE = JK and FE = MK. Since FD = MJ we can stay the equation 2x = 8. Solving this equation we get x=4 4 F M 2X 8 D E 12 J Now, find the x value, if triangles. 13 DEF and F K KJM are congruent M X+7 16 E K D 10 13 J 5. Given: ABC and RST, with AB = RS, BC = ST, and AC = RT. Copy and complete each row of angle measures. (IA) <A 70° 100° <B 80° <C <R <S <T 20° 52° 63° x° y° SAS Postulate 5 If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. 6. Given: ⃗⃗⃗⃗⃗⃗⃗ 𝒁𝑾 bisects <XZY XZ = YZ Prove: XWZ ≅ YWZ Z X 4. W STATEMENTS 1. <XZW = < WZY 2. ZW = ZW 3. XZ = YZ XWZ ≅ YWZ Y REASONS 1. Given: ⃗⃗⃗⃗⃗⃗⃗ 𝒁𝑾 bisects <XZY 2. Reflexive Property 3. Given 4. SAS Postulate Try 7. Complete the proof by supplying the reasons. A D 6 C X B Given: X is the midpoint of ̅̅̅̅ 𝑨𝑩 and ̅̅̅̅ 𝑪𝑫 Prove: 4. AXC ≅ BXD STATEMENTS 1. CX = DX AX=BX 2. AX=BX 3. <AXC = <BXD AXC ≅ BXD REASONS ASA Postulate If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. 7 Name the side included between the two angles. X C 5 R K 8 B 6 9 A 4 1. <R and <K 2. <X and <R 7 3. <5 and <6 <7 and <8 D 8. Given: M is the midpoint of ̅̅̅̅ 𝒀𝒁. ⃗⃗⃗⃗⃗⃗⃗ 𝑴𝒀 bisects <OMV Prove: YOM ≅ ZUM O Y U 1 M 2 Z 3 V 8 STATEMENTS 1. <1= <3 2. <2 = <3 3. <1 = <2 4. YM = ZM 5. < Y = < Z 6. YOM ≅ ZUM REASONS 1. Given: ⃗⃗⃗⃗⃗⃗⃗ 𝑴𝒀 bisects <OMV 2. Vertical angles are congruent 3. Substitution Postulate 4. Given: M is the midpoint of⃗⃗⃗⃗⃗⃗ 𝒀𝒁 5. Given 6. ASA Postulate. AAS Theorem If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. HL Theorem If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle, the triangles are congruent. 9 9. Given: <B and <X are right angles. BY = AX Prove: ABY ≅ YXA X A STATEMENTS 1. <B and <X are right angles 2. AY = AY 3. BY = AX 4. ABY ≅ YXA Y B REASONS 1. Given 2. Reflexive Property 3. Given 4. HL 10 STRATEGIES AND PRACTICES PROVIDE HINTS AND FEEDBACK INTERACTIV E PARTICIPATION HAVING STUDENTS TEACH WHAT THEY LEARNED TO SOMEONE ELSE BLOOM TAXONOMY (CONGNITIVE DOMAIN) 1. KNOWLEDGE Name 2. COMPREHENSION Recognize 3. APPLICATION Sketch 4. ANALYSIS Dif ferentiate TEACHING RESOURCES & MATERIALS DOCUMENT CAMERA WHITE BOARD MARKERS COMPASS PROTRACTOR CARDBOARD PAPER RULER 11 12