1. 5.1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004

2. Objectives: Use properties of perpendicular bisectors Use properties of angle bisectors to identify equal distances.

3. Theorem 5.2 Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA = CB.

4. Theorem 5.3: Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB.

5. Assignment page 267-268 #1-25 All

6. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given Given: CP is perpendicular to AB. Prove: CA ≅CB

7. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector Given: CP is perpendicular to AB. Prove: CA ≅CB

8. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector 3.Given Given: CP is perpendicular to AB. Prove: CA ≅CB

9. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector 3.Given 4.Reflexive Prop. Congruence. Given: CP is perpendicular to AB. Prove: CA ≅CB

10. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector 3.Given 4.Reflexive Prop. Congruence. 5.Definition right angle Given: CP is perpendicular to AB. Prove: CA ≅CB

11. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector 3.Given 4.Reflexive Prop. Congruence. 5.Definition right angle 6.SAS Congruence Given: CP is perpendicular to AB. Prove: CA ≅CB

12. Statements: 1.CP is perpendicular bisector of AB. 2.CP  AB 3.AP ≅ BP 4. CP ≅ CP 5.  CPB ≅  CPA 6.∆APC ≅ ∆BPC 7.CA ≅ CB Reasons: 1.Given 2.Definition of Perpendicular bisector 3.Given 4.Reflexive Prop. Congruence. 5.Definition right angle 6.SAS Congruence 7.CPCTC Given: CP is perpendicular to AB. Prove: CA ≅CB

13. Theorem 5.4 Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m  BAD = m  CAD, then DB = DC

14. Theorem 5.4 Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If DB = DC, then m  BAD = m  CAD.

15. SOLUTION: