5 1 Perpendiculars and Bisectors Geometry Mrs Spitz

5. 1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004

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

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.

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.

Assignment • page 267 -268 #1 -25 All

Given: CP is perpendicular to AB. Prove: CA≅CB 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 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 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 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 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 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 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

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

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.

SOLUTION: