Section 5-2Bisectors in Triangles

Vocabulary

• Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Theorems

• Perpendicular Bisector Theorem- If a point lies on the perpendicular bisector of a line segment, then it is an equal distance away from both endpoints of the line segment.

• Angle Bisector Theorem- If a point lies on the angle bisector of an angle, then it is an equal distance away from both sides of the angle.

Converse of the Theorems

• Converse of the Perpendicular Bisector Theorem- If a point is an equal distance away from the endpoints of a line segment, then it lies on the perpendicular bisector of the line segment.

• Converse of the Angle Bisector Theorem- If a point in the interior of an angle is an equal distance away from both sides of the angle, then it lies on the angle bisector of the angle.

Perpendicular Bisector Theorem

Angle Bisector Theorem

Proof of Perpendicular Bisector Theorem

Statement Reason

BD is the bisector of AC┴ Given

∠ABD and CBD are right angles∠ Definition of perpendicular

∠ABD ∠ CBD ∠ All right angles are congruent

DB DB≅ Reflexive Property of Congruency

AB CB≅ Definition of bisector

∆ABD ∆CBD≅ SAS

AD CD≅ CPCTC

Proof of Angle Bisector TheoremStatement Reason

AD is the angle bisector of CAB∠ Given

CD is to AC┴ By construction

DB is to AB┴ By construction

∠ACB and ABD are right angles∠ Definition of perpendicular

∠ACB ∠ ABD ∠ All right angles are congruent

∠CAD BAD≅ ∠ Definition of angle bisector

AD AD≅ Reflexive Property pf Congruency

∆CAD ∆BAD≅ AAS

CD BD≅ CPCTC

Practice Problem

• Given: BE is the perpendicular bisector of AC, AED CEF, ∠ ≅ ∠DE FE.≅

• Prove: DAE FCE∠ ≅ ∠Answer on next slide⫸

Solution to Practice ProblemStatement Reason

BE is ┴ bisector of AC Given

AE CE≅ Perpendicular Bisector Theorem

DE FE≅ Given

∠AED CEF≅ ∠ Given

∆ADE ∆CFE≅ SAS

∠DAE FCE≅ ∠ CPCTC

Practice Problem 2

• Find the value of X and YAnswer on next slide⫸

Solution to Practice Problem 2

• Answer: X = -3, Y = 12

Extra Resources

• http://www.youtube.com/watch?v=oskp0T8aZJw (very weird jumpy guy explaining the perpendicular bisector theorem)

• http://www.youtube.com/watch?v=9k8QMHIFwOk&list=PL668AB35C6885A036&index=35 (same weird guy explaining the angle bisector theorem)