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9 Deductive Geometry 9.1 Introduction to Deductive Reasoning and Proofs 9.2 Deductive Proofs Related to Lines and Triangles 9.3 Deductive Proofs Related to Congruent and Isosceles Triangles 9.4 Deductive Proofs Related to Similar Triangles 9.1 Introduction to Deductive Reasoning and Proofs A. What is Deductive Reasoning? 9.1 Introduction to Deductive Reasoning and Proofs A. What is Deductive Reasoning? 9.1 Introduction to Deductive Reasoning and Proofs B. Euclid and ‘Elements’ 9.1 Introduction to Deductive Reasoning and Proofs B. Euclid and ‘Elements’ 9.1 Introduction to Deductive Reasoning and Proofs C. Deductive Proofs of Theorems 9.1 Introduction to Deductive Reasoning and Proofs C. Deductive Proofs of Theorems 9.2 Deductive Proofs Related to Lines and Triangles A. Angles Related to Intersecting Lines 9.2 Deductive Proofs Related to Lines and Triangles A. Angles Related to Intersecting Lines 9.2 Deductive Proofs Related to Lines and Triangles A. Angles Related to Intersecting Lines 9 Deductive Geometry Example 1T In the figure, AOB, COD and EOF are straight lines. AOC = 90. Prove that a + b = 90. Solution: AOE = BOF = b AOC + AOE + DOE = 180 90 + b + a = 180 a + b = 90 vert. opp. s adj. s on st. line 9.2 Deductive Proofs Related to Lines and Triangles A. Angles Related to Intersecting Lines 9 Deductive Geometry Example 2T In the figure, AFB is a straight line. If CFB = AFD, prove that CFD is a straight line. Solution: AFD + DFB = 180 CFB = AFD Consider CFB + DFB AFD + DFB 180 ∴ CFD is a straight line. adj. s on st. line given given adj. s supp. 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9 Deductive Geometry Example 3T In the figure, AB // DE, BAC = 150 and EDC = 120. Prove that AC CD. Solution: Construct a line CF such that AB // CF // DE. a + 150 = 180 a = 30 b + 120 = 180 b = 60 Consider ACD = a + b = 30 + 60 = 90 ∴ AC CD int. s, AB // CF int. s, CF // DE 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9.2 Deductive Proofs Related to Lines and Triangles B. Angles Related to Parallel Lines 9 Deductive Geometry Example 4T In the figure, AGHB, CGD and EHF are straight lines. Prove that CD // EF. Solution: DGH AGC x ∵ DGH FHB x ∴ CD // EF vert. opp. s corr. s equal 9 Deductive Geometry Example 5T In the figure, BAC = 25, reflex ACD = 305 and CDE = 30. Prove that AB // DE. Solution: Construct a line FC such that AB // FC. a BAC alt. s, AB // FC 25 b 360 – 305 – a s at a pt. 360 – 305 – 25 30 ∵ CDE b 30 ∴ FC // DE alt. s equal ∴ AB // DE 9.2 Deductive Proofs Related to Lines and Triangles C. Angles Related to Triangles 9.2 Deductive Proofs Related to Lines and Triangles C. Angles Related to Triangles 9 Deductive Geometry Example 6T In the figure, EDB is a straight line. ABD = 100, EDC = 120 and DCB = 40. Prove that ABC is a straight line. Solution: In BCD, BCD + CBD CDE 40 + CBD 120 CBD 80 Consider ABC CBD + ABD 80 + 100 180 ∴ ABC is a straight line. ext. of adj. s supp. 9.3 Deductive Proofs Related to Congruent and Isosceles Triangles A. Congruent Triangles 9 Deductive Geometry Example 7T In the figure, AD = AB and CD = CB. (a) Prove that ABC ADC. (b) Prove that DCA = BCA. Solution: (a) In ABC and ADC, AB = AD CB = CD AC = AC ∴ ABC ADC (b) ∴ DCA = BCA given given common side SSS corr. s, s 9 Deductive Geometry Example 8T In the figure, BAE = BCD and AB = BC. (a) Prove that ABE CBD. (b) Prove that DF = EF. Solution: (a) In ABE and CBD, ABE = CBD AB = CB BAE = BCD ∴ ABE CBD (b) ∵ AB = CB and BD = BE ∴ AB – BD = CB – BE AD = CE common given given ASA given corr. sides, s 9 Deductive Geometry Example 8T In the figure, BAE = BCD and AB = BC. (a) Prove that ABE CBD. (b) Prove that DF = EF. Solution: In ADF and CEF, AFD = CFE DAF = ECF AD CE ∴ ADF CEF ∴ DF = EF vert. opp. s given proved AAS corr. sides, s 9.3 Deductive Proofs Related to Congruent and Isosceles Triangles B. Isosceles Triangles 9 Deductive Geometry Example 9T In the figure, AB = AC and ACD = DCB. Prove that ADC = 3ACD. Solution: In ABC, ∵ ACD = DCB ∴ ACB 2ACD ∴ ABC = ACB = 2ACD In BCD, ADC = ABC + DCB = 2ACD + ACD = 3ACD given base s, isos. ext. of 9.3 Deductive Proofs Related to Congruent and Isosceles Triangles B. Isosceles Triangles 9 Deductive Geometry Example 10T In the figure, ABC is a straight line. AB = DB and ADC = 90. Prove that BCD is an isosceles triangle. Solution: In ABD, ∵ ∴ ∵ ∴ AB = DB BAD = BDA BDC + BDA 90 BDC 90 – BDA = 90 – BAD In ACD, ACD + CAD + ADC = 180 BCD + BAD + 90 = 180 ∴ BCD 90 – BAD ∵ BCD = BDC ∴ BC = BD ∴ BCD is an isosceles triangle. given base s, isos. given sum of proved sides opp. eq. s 9 Deductive Geometry Example 11T In the figure, BD = CD and ADB = ADC. (a) Prove that ADB ADC. (b) Prove that ABC = ACB. Solution: (a) In ADB and ADC, AD = AD BD = CD ADB = ADC ADB ADC (b) In ABC, ∵ AB AC ∴ ABC ACB common side given given SAS corr. sides, s base s, isos. 9.4 Deductive Proofs Related to Similar Triangles 9 Deductive Geometry Example 12T In the figure, ABC and AED are straight lines. AEB = ADC. Prove that ABE ~ ACD. Solution: ∵ AEB = ADC ∴ BE // CD In ABE and ACD, BAE = CAD ABE = ACD AEB ADC ∴ ABE ~ ACD given corr. s equal common corr. s, BE // CD given AAA 9 Deductive Geometry Example 13T In the figure, ABD and CBE are straight lines. Prove that ABE ~ DBC. Solution: In ABE and DBC, AB 2 1 DB 6 3 BE 3 1 BC 9 3 ABE = DBC ∴ ABE ~ DBC vert. opp. s ratio of 2 sides, inc. 9 Deductive Geometry Example 14T In the figure, ACB = 47. AB = 4, BC = 5, CD = 1, AD = DE = 4 and AE = 3.2. Find BAE. Solution: In ABC and AED, AB 4 5 AE 3.2 4 BC 5 ED 4 AC 4 1 5 AD 4 4 AB BC AC 5 ∵ = = = AE ED AD 4 ∴ ABC ~ AED 3 sides proportional 9 Deductive Geometry Example 14T In the figure, ACB = 47. AB = 4, BC = 5, CD = 1, AD = DE = 4 and AE = 3.2. Find BAE. Solution: In ABC, ∵ AC = 4 + 1 = 5 = BC ∴ ABC = BAC ABC + BAC + ACB = 180 BAC + BAC + 47 = 180 BAC = 66.5 ∴ EAD = BAC = 66.5 BAE = BAC + EAD = 66.5 + 66.5 = 133 base s, isos. sum of corr. s, ~ s