1. Splash Screen

2. We learned earlier that a segment bisector is any line, segment, or plane that intersects a segment at its midpoint. If a bisector is also perpendicular to the segment, it is called a perpendicular bisector.
Concept

3. Concept

4. Use the Perpendicular Bisector Theorems
A. Find BC.
Example 1

5. Use the Perpendicular Bisector Theorems
B. Find XY.
Example 1

6. Use the Perpendicular Bisector Theorems
C. Find PQ.
Example 1

7. When three or more lines intersect at a common point, the lines are called concurrent lines. The point where concurrent lines intersect is called the point of concurrency.
A triangle has three sides, so it also has three perpendicular bisectors. These bisectors are concurrent lines. The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle.
Concept

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

9. Concept

10. Use the Circumcenter Theorem
GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? C
Example 2

11. The angle bisector can be a line, segment, or ray.
We learned earlier that an angle bisector divides an angle into two congruent angles.
The angle bisector can be a line, segment, or ray.
Concept

12. Concept

13. Use the Angle Bisector Theorems
A. Find DB.
Example 3

14. Use the Angle Bisector Theorems
B. Find mWYZ.
Example 3

15. Use the Angle Bisector Theorems
C. Find QS.
Example 3

16. The angle bisectors of a triangle are concurrent, and their point of concurrency is called the incenter of a triangle.
Concept

17. A. Find ST if S is the incenter of ΔMNP.
Use the Incenter Theorem
A. Find ST if S is the incenter of ΔMNP.
Example 4

18. B. Find mSPU if S is the incenter of ΔMNP.
Use the Incenter Theorem
B. Find mSPU if S is the incenter of ΔMNP.
Example 4

19. End of the Lesson

20. Five-Minute Check (over Chapter 4) CCSS Then/Now New Vocabulary
Theorems: Perpendicular Bisectors
Example 1: Use the Perpendicular Bisector Theorems
Theorem 5.3: Circumcenter Theorem
Proof: Circumcenter Theorem
Example 2: Real-World Example: Use the Circumcenter Theorem
Theorems: Angle Bisectors
Example 3: Use the Angle Bisector Theorems
Theorem 5.6: Incenter Theorem
Example 4: Use the Incenter Theorem
Lesson Menu

21. Classify the triangle. A. scalene B. isosceles C. equilateral
5-Minute Check 1

22. Find x if mA = 10x + 15, mB = 8x – 18, and mC = 12x + 3.
5-Minute Check 2

23. Name the corresponding congruent sides if ΔRST  ΔUVW.
A. R  V, S  W, T  U
B. R  W, S  U, T  V
C. R  U, S  V, T  W
D. R  U, S  W, T  V
5-Minute Check 3

24. Name the corresponding congruent sides if ΔLMN  ΔOPQ.B. C. D. ,
5-Minute Check 4

25. Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.B
C. 7
D. 4.5
5-Minute Check 5

26. ΔABC has vertices A(–5, 3) and B(4, 6)
ΔABC has vertices A(–5, 3) and B(4, 6). What are the coordinates for point C if ΔABC is an isosceles triangle with vertex angle A?
A. (–3, –6)
B. (4, 0)
C. (–2, 11)
D. (4, –3)
5-Minute Check 6

27. G.CO.10 Prove theorems about triangles.
Content Standards
G.CO.10 Prove theorems about triangles.
G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
CCSS

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