1. 1. Prove that the three angle bisectors of a triangle concur. C AB D F E I § 4.1

2. 2. Prove that the perpendicular bisectors of a triangle concur. C AB O l 3l 3 l 2l 2 l 1l 1

3. 3. Prove that the altitudes of a triangle concur. R T C B Q P A O V U Note that the altitudes of ∆ABC are the perpendicular bisectors of the sides of ∆PQR and using the previous problem the perpendicular bisectors concur.

4. 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 C D B E F G StatementReason 1. Let E be the midpoint of AC.Construction 3.  BEA =  CEF Vertical angles 4. ∆AEB = ∆ECFSAS 5. m  A = m  ECF CPCTE 2. Choose F on BE so that BE = EFConstruction. 6. m  A + m  FCG = m  ECF + m  FCG Arithmetic 7. m  ACG = m  ECF + m  FCG Angle Addition 8. m  ACG = m  A + m  FCG CPCTE & Substitution 9. m  ACG > m  A Arithmetic 10.  ACG >  A Angle measure postulate

5. 6. Given: AD bisects  CAB and CA = CD. Prove: CD parallel to AB.

6. 7. Segments AB and CD bisect each other at E. Prove that AC is parallel to BD. A E D C B StatementReason 1. AE = EB & CE = ED1. 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.

7. 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. StatementReason 1.  A =  B. 1. Given 2.  B =  C. 2. Vertical angles 3.  A =  C. 3. Transitive Given: l and m cut by transversal m and  A =  B. A C B l t m Prove:  A =  C.

8. 9. Given two lines cut by a transversal. If a pair of corresponding angles are congruent, prove that the lines are parallel. StatementReason 1.  A =  C. 1. Previous problem 2. l and m parallel.2. Definition of parallel Given: l and m cut by transversal m and  A =  B. A C B l t m Prove: l and m parallel.

9. 11. Given triangle ABC with AC = BC and DC = EC, and  EDC =  EBA, prove DE is parallel to AB. StatementReason 1. AC = BC and DC = EC1. 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 Given: AC = BC and DC = EC, and  EDC =  EBA Prove: DE is parallel to AB.