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§ 4.1 C 1. Prove that the three angle bisectors of a triangle concur. E Given: CAD BAD, ABE CBE AND ACF BCF A Prove: AD, BE AND CF concurr. Statement Reason 1. Let AD ∩ BE = I Two lines intersect. 2. I is in the interior of A 3. I is in the interior of B On the bisector On the bisector 4. I is in the interior of C 5. I is equidistant from AC and AB 6. I is equidistant from AB and BC From 2 & 3 Def. angle bisector Def. angle bisector 7. So I is equidistant from AC and BC Transitive property I is on CF and CF is a bisector of C Def of bisector. I F D B C 2. Prove that the perpendicular bisectors O of a triangle concur. Given: CAD BAD, ABE CBE AND ACF BCF A Prove: AD, BE AND CF concurr. Statement Reason 1. Let l 1 ∩ l 2 = O Two lines intersect. 2. BO = CO 3. AO = CO O is on the bisector of AB O is on the bisector of AC 4. BO = AO 5. O is on l 3 Transitive property Def. bisector l 1 , l 2 and l 3 concur at O l1 l2 B l3 Q U C T 3. Prove that the altitudes of a triangle concur. P O A B V R Note that the altitudes of ∆ABC are the perpendicular bisectors of the sides of ∆PQR and using the previous problem the perpendicular bisectors concur. B 4. Complete the proof that the exterior angle of a triangle is greater than each of its remote interior angles. Given: A – C – D and ∆ ABC Prove ACG > A A E D C F G Statement Reason 1. Let E be the midpoint of AC. 2. Choose F on BE so that BE = EF Construction Construction. 3. BEA = CEF 4. ∆AEB = ∆ECF Vertical angles SAS CPCTE Arithmetic Angle Addition CPCTE & Substitution Arithmetic Angle measure postulate 5. m A = m ECF 6. m A + m FCG = mECF + m FCG 7. mACG = mECF + m FCG 8. mACG = mA + m FCG 9. mACG > mA 10. ACG > A 6. Given: AD bisects CAB and CA = CD. Prove: CD parallel to AB. Given: AD bisects CAB and CA = CD. Prove: CD parallel to AB. Statement Reason 1. CAD = DAB Given AD bisector 2. CA = CD 3. CAD = CDA 4. CDA = DAB Given s opposite = sides = Transitive property 5. CD parallel to AB s in 4 are alternate interior B 7. Segments AB and CD bisect each other at E. Prove that AC is parallel to BD. E C D A Statement Reason 1. AE = EB & CE = ED 1. Given 2. AEC = BED 2. Vertical angles 3. AEC = BED 3. SAS 4. CAE = DBE 4. CPCTE 5. AC parallel to BD 5. s in 4 are alternate interior angles. 8. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, prove that a pair of alternate interior angles are congruent. t B C A Given: l and m cut by transversal m and A = B. Prove: A = C. Statement Reason 1. A = B. 1. Given 2. B = C. 2. Vertical angles 3. A = C. 3. Transitive l m 9. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, prove that the lines are parallel. t B C A Given: l and m cut by transversal m and A = B. Prove: l and m parallel. Statement Reason 1. A = C. 1. Previous problem 2. l and m parallel. 2. Definition of parallel l m 11. Given triangle ABC with AC = BC and DC = EC, and EDC = EBA, prove DE is parallel to AB. Given: AC = BC and DC = EC, and EDC = EBA Prove: DE is parallel to AB. Statement Reason 1. AC = BC and DC = EC 1. Given 2. ABC and AEC isosceles. 2. Definition of isosceles 3. EDC = EBA. 3. Given. 4. EDC = DEC. 4. Base angles. 5. DEC = EBA. 5. Substitution of 3 into 4. 6. DEC & EBA are corresponding s. 6. Definition. 7. DE is parallel to AB. 7. Corresponding s. equal. 5 & 6