5.6 ESSENTIAL OBJECTIVES: Use angle bisectors and perpendicular bisectors in solving triangles. How do you apply the angle bisector and perpendicular bisector concept in solving for missing parts of a triangle?

Angle Bisector A segment that bisects, or splits an angle into 2 congruent angles.

Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Reflexive Prop. of Congruence 3. WU  WU Angle Bisector Theorem 4. Example 1 Use the Angle Bisector Theorem ∆TWU  ∆VWU. UW bisects TUV. SOLUTION Statements Reasons 1. Given UW bisects TUV. . Reflexive Prop. of Congruence 3. WU  WU Angle Bisector Theorem 4. WV  WT HL Congruence Theorem 5. ∆TWU  ∆VWU

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 1. Find FH.

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 1. Find FH.

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 1. Find FH.

1. Find FH. Checkpoint Use Angle Bisectors and Perpendicular Bisectors

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 2. Find MK.

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 2. Find MK.

Perpendicular Bisector A segment that bisects, or splits an angle into 2 congruent angles.

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 3. Find EF.

Checkpoint Use Angle Bisectors and Perpendicular Bisectors 3. Find EF.

Use the diagram to find AB. Example 2 Use Perpendicular Bisectors Use the diagram to find AB. SOLUTION In the diagram, AC is the perpendicular bisector of DB. 8x = 5x +12 By the Perpendicular Bisector Theorem, AB = AD. 3x = 12 Subtract 5x from each side. 3 3x 12 = Divide each side by 3. x = 4 Simplify. You are asked to find AB, not just the value of x. ANSWER AB = 8x = 8 · 4 = 32

will be equidistant from each store. Example 4 Use Intersecting Bisectors of a Triangle A company plans to build a warehouse that is equidistant from each of its three stores, A, B, and C. Where should the warehouse be built? SOLUTION Think of the stores as the vertices of a triangle. The point where the perpendicular bisectors intersect will be equidistant from each store. Trace the location of the stores on a piece of paper. Connect the points of the locations to form ∆ABC. 1. 16

Draw the perpendicular bisectors Example 4 Use Intersecting Bisectors of a Triangle 2. Draw the perpendicular bisectors of AB, BC, and CA. Label the intersection of the bisectors P. ANSWER Because P is equidistant from each vertex of ∆ABC, the warehouse should be built near location P. 17

Review

1. Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show that the triangles are congruent. KL  NM, KLM  NML ANSWER SAS Congruence Postulate

2. Use the information in the diagram to show that the given statement is true. HJ  KJ ANSWER It is given that GH  GK, and GJ  GJ by the Reflexive Property of Congruence. ∆GHJ and ∆GKJ are right triangles by definition. By the HL Congruence Theorem, ∆GHJ  ∆GKJ. Therefore, HJ  KJ because corresponding parts of congruent triangles are congruent.

Hw 5.6A & 5.6B (4-9)