5.1 Perpendiculars and Bisectors

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. 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. Theorem 5.2 Perpendicular Bisector Theorem

4. 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. Theorem 5.3: Converse of the Perpendicular Bisector Theorem

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

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

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

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

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

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

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

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

13. 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 Theorem 5.4 Angle Bisector Theorem

14. 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. Theorem 5.4 Angle Bisector Theorem

15. SOLUTION: