1. **Homework: #1-6 from 5.1 worksheet**
Agenda 11/8
Pass out chapter 5 workbooks
Start Section 5.1
**Homework: #1-6 from 5.1 worksheet**

2. Section 5.1 Notes Part 1: Bisectors of Triangles
EQ: What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

3. Perpendicular Bisector
Equidistant
 Equal distance
Segment Bisector
A point, segment, line or plane that divides a line segment into two equal parts.
Perpendicular Bisector
If a bisector is also perpendicular to the segment, it is called a perpendicular bisector

4. If 𝐶𝐷 is a perpendicular bisector of 𝐴𝐵 , then AC = BC
Perpendicular Bisector Theorem
 If 𝐶𝐷 is a perpendicular bisector of 𝐴𝐵 , then AC = BC
Converse of the Perpendicular Bisector Theorem
If AE = BE, then E lies on 𝐶𝐷 , the perpendicular bisector of 𝐴𝐵

5. Example 1:
a) Find the length of BC.
𝐴𝐶 = 𝐵𝐶
𝐵𝐶 =8.5

6. Example 1:
b) Find the length of XY.
𝑍𝑌 = 𝑋𝑌
𝑋𝑌 =6

7. Example 1:
c) Find the length of PQ.
𝑃𝑄 = 𝑅𝑄
3𝑥+1=5𝑥−3
𝑥=2
𝑃𝑄 =7

8. Try the you try first before you look at the answer!

9. YOU TRY!
1. RS 2. EG 3. AD
𝑅𝑆 =6.8
𝐸𝐺 =19
3𝑥+14=5𝑥
𝑥=7
𝐴𝐷 =35

10. If 𝐹𝐷 ⊥ 𝐵𝐷 , 𝐹𝐸 ⊥ 𝐵𝐸 , and DF = FE, then 𝐵𝐹 bisects ∠𝐷𝐵𝐸
Angle Bisector Theorem
 If 𝐵𝐹 bisects ∠𝐷𝐵𝐸, 𝐹𝐷 ⊥ 𝐵𝐹 , & 𝐹𝐸 ⊥ 𝐵𝐸 , then DF = FE
Converse of the Angle Bisector Theorem
 If 𝐹𝐷 ⊥ 𝐵𝐷 , 𝐹𝐸 ⊥ 𝐵𝐸 , and DF = FE, then 𝐵𝐹 bisects ∠𝐷𝐵𝐸

11. Example 2:
a) Find the length of DB.
𝐷𝐵 =5

12. Example 2:
b) Find m∠WYZ.
∠𝑊𝑌𝑍 = 68°

13. Example 2:
c) Find the length of QS.
4𝑥−1=3𝑥+2
𝑥=3
𝑄𝑆 =11

14. YOU TRY!
Find each measure.
𝑚∠𝐺𝐹𝐽 
 2. RS
∠𝐺𝐹𝐽 = 42°
5𝑥=6𝑥−5
𝑥=5
𝑅𝑆 =25

15. Section 5.1 Notes Part 2: Bisectors of Triangles
EQ: What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

16. When three or more lines intersect at a common point
Concurrent Lines
When three or more lines intersect at a common point
Point of Concurrency   
The point where concurrent lines intersect
Circumcenter
The point of concurrency of the perpendicular bisectors
Circumcenter Theorem
If P is the circumcenter of ∆ABC, then PB = PA = PC

17. Example 4
A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?
No, the circumcenter is outside of the garden.

18. If P is the incenter of ∆ABC, then PD = PE = PF
Incenter
The angle bisectors of a triangle are concurrent, and their point of concurrency is the incenter
Incenter Theorem
If P is the incenter of ∆ABC, then PD = PE = PF 

19. Example 5 Use the diagram to the right.
Find ST if S is the incenter of ΔMNP.
𝑎 2 + 𝑏 2 = 𝑐 2
𝑆𝑈 = 10 2
𝑆𝑈 2 =36
𝑆𝑈=6
𝑆𝑈=𝑆𝑅=𝑆𝑇
Therefore 𝑆𝑇=6

20. b) Find mSPU if S is the incenter of ΔMNP.
𝑚∠𝑁𝑀𝑃=31+31=62°
𝑚∠𝑀𝑁𝑃=180−56−62=62°
𝑚∠𝑆𝑃𝑈= =31°

21. YOU TRY!
In the figure shown, ND= 5 x – 1 and NE = 2 x Find NF
ND = NE
5𝑥−1=2𝑥+11
𝑥=4
2(4) = 11
NE = 11
ND = NE = NF
NF = 11