Section 5-1 Bisectors in Triangles

Then Now Objectives You used segment and angle bisectors. Identify and use perpendicular bisectors in triangles. Identify and use angle bisectors in triangles.

Common Core State Standards Content Standards G.CO.10 – Prove theorems about triangles. G.MG.3 – Apply geometric methods to solve problems. Mathematical Practices 1) Make sense of problems and persevere in solving them. 3) Construct viable arguments and critique the reasoning of others. Common Core State Standards

Perpendicular Bisector: A bisector that is also perpendicular to a segment. Vocabulary

Perpendicular Bisectors

If m is the perpendicular bisector of XZ and WZ = 14.9, find WX. Example 1

If WX = 25. 3, YZ = 22. 4, and WZ = 25. 3, find XY If WX = 25.3, YZ = 22.4, and WZ = 25.3, find XY. If m is the perpendicular bisector of XZ, WX = 4a – 15, and WZ = a + 12, find WX. Example 1

Concurrent Lines: Three or more lines that intersect at a common point Concurrent Lines: Three or more lines that intersect at a common point. Point of Concurrency: The point where concurrent lines intersect. Vocabulary

Circumcenter: The point of concurrency of the perpendicular bisectors in a triangle. Vocabulary

The circumcenter of a triangle can be on the interior, exterior, or side of a triangle.

Circumcenter Theorem

Point P is the circumcenter of ∆ABC Point P is the circumcenter of ∆ABC. List any segment(s) congruent to each segment. a) 𝐵𝑅 b) 𝐶𝑆 c) 𝐵𝑃 Example 2

Point L is the circumcenter of ∆KTB Point L is the circumcenter of ∆KTB. List any segment(s) congruent to each segment. a) 𝐵𝑁 b) 𝐵𝐿 c) 𝐾𝐻 Example 2

An angle bisector divides an angle into two congruent angles. Remember!

Angle Bisectors

If m∠BAC = 40, m∠DAC = 40, and DC = 10, find BC. Example 3

If m∠BAC = 38, BC = 5, and DC = 5, find m∠DAB If m∠BAC = 38, BC = 5, and DC = 5, find m∠DAB. If 𝐴𝐶 bisects ∠DAB, BC = 4x + 8, and DC = 9x – 7, find BC. Example 3

Incenter: The point of concurrency of the angle bisectors of a triangle. *The incenter ALWAYS lies in the INTERIOR of a ∆! Vocabulary

Incenter Theorem

Example 4 Point P is the incenter of ∆XYZ. Find each measure. PK m∠LZP LP Example 4

Example 4 Point J is the incenter of ∆ABC. Find each measure. JD m∠JCF JE m∠BAJ Example 4

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