1. Section 5-1 Bisectors in Triangles

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

3. 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

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

5. Perpendicular Bisectors

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

7. 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

8. 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

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

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

11. Circumcenter Theorem

12. 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

13. 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

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

15. Angle Bisectors

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

17. 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

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

19. Incenter Theorem

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

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

22. p.329 #1 – 29 odd, skip #15
Homework