1. Section 5-2 Bisectors in Triangles

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

3. 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.

4. 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.

5. Perpendicular Bisector Theorem

6. Angle Bisector Theorem

7. Proof of Perpendicular Bisector Theorem StatementReason 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

8. Proof of Angle Bisector Theorem StatementReason 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

9. Practice Problem Given: BE is the perpendicular bisector of AC, ∠AED ≅ ∠CEF, DE ≅ FE. Prove: ∠DAE ≅ ∠FCE Answer on next slide⫸

10. Solution to Practice Problem StatementReason BE is ┴ bisector of ACGiven AE ≅ CE Perpendicular Bisector Theorem DE ≅ FE Given ∠AED ≅ ∠CEF Given ∆ADE ≅ ∆CFE SAS ∠DAE ≅ ∠FCE CPCTC

11. Practice Problem 2 Find the value of X and Y Answer on next slide⫸

12. Solution to Practice Problem 2 Answer: X = -3, Y = 12

13. Extra Resources http://www.youtube.com/watch?v=oskp0T8aZJw (very weird jumpy guy explaining the perpendicular bisector theorem) http://www.youtube.com/watch?v=oskp0T8aZJw http://www.youtube.com/watch?v=9k8QMHIFwOk&list=PL66 8AB35C6885A036&index=35 (same weird guy explaining the angle bisector theorem) http://www.youtube.com/watch?v=9k8QMHIFwOk&list=PL66 8AB35C6885A036&index=35