1. Five-Minute Check (over Chapter 4) Mathematical Practices 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

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

3. 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

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

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

6. 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

7. Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
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). MP

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

9. perpendicular bisector
concurrent lines
point of concurrency
circumcenter
incenter
Vocabulary

10. BC = AC Perpendicular Bisector Theorem BC = 8.5 Substitution
Use the Perpendicular Bisector Theorems
A. Find BC.
BC = AC Perpendicular Bisector Theorem
BC = 8.5 Substitution
Answer: 8.5
Example 1

11. B. Find XY. Answer: 6 Use the Perpendicular Bisector Theorems
Example 1

12. PQ = RQ Perpendicular Bisector Theorem 3x + 1 = 5x – 3 Substitution
Use the Perpendicular Bisector Theorems
C. Find PQ.
PQ = RQ Perpendicular Bisector Theorem
3x + 1 = 5x – 3 Substitution
1 = 2x – 3 Subtract 3x from each side.
4 = 2x Add 3 to each side.
2 = x Divide each side by 2.
So, PQ = 3(2) + 1 = 7.
Answer: 7
Example 1

13. A. Find NO.
A. 4.6
B. 9.2
C. 18.4
D. 36.8
Example 1

14. B. Find TU.
A. 2
B. 4
C. 8
D. 16
Example 1

15. C. Find EH.
A. 8
B. 12
C. 16
D. 20
Example 1

16. Concept

17. Concept

18. 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?
By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.
Example 2

19. Use the Circumcenter Theorem
Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C
Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.
Example 2

20. BILLIARDS A triangle used to rack pool balls is shown
BILLIARDS A triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle?
A. No, the circumcenter of an acute triangle is found in the exterior of the triangle.
B. Yes, circumcenter of an acute triangle is found in the interior of the triangle.
Example 2

21. Concept

22. DB = DC Angle Bisector Theorem DB = 5 Substitution
Use the Angle Bisector Theorems
A. Find DB.
DB = DC Angle Bisector Theorem
DB = 5 Substitution
Answer: DB = 5
Example 3

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

24. WYZ  XYW Definition of angle bisector
Use the Angle Bisector Theorems
WYZ  XYW Definition of angle bisector
mWYZ = mXYW Definition of congruent angles
mWYZ = 28 Substitution
Answer: mWYZ = 28
Example 3

25. QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution
Use the Angle Bisector Theorems
C. Find QS.
QS = SR Angle Bisector Theorem
4x – 1 = 3x + 2 Substitution
x – 1 = 2 Subtract 3x from each side.
x = 3 Add 1 to each side.
Answer: So, QS = 4(3) – 1 or 11.
Example 3

26. A. Find the measure of SR.
A. 22
B. 5.5
C. 11
D. 2.25
Example 3

27. B. Find the measure of HFI.
C. 15
D. 30
Example 3

28. C. Find the measure of UV.
A. 7
B. 14
C. 19
D. 25
Example 3

29. Concept

30. A. Find SU if S is the incenter of ΔMNP.
Use the Incenter Theorem
A. Find SU if S is the incenter of ΔMNP.
By the Incenter Theorem, since S is equidistant from the sides of ΔMNP, ST = SU.
Find SU by using the Pythagorean Theorem.
a2 + b2 = c2 Pythagorean Theorem
82 + SU2 = 102 Substitution
64 + SU2 = = 64, 102 = 100
Example 4

31. SU2 = 36 Subtract 64 from each side.
Use the Incenter Theorem
SU2 = 36 Subtract 64 from each side.
SU = ±6 Take the square root of each side.
Since length cannot be negative, use only the positive square root, 6. Since ST = SU, SU = 6.
Answer: SU = 6
Example 4

32. 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.
Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, mTNU = 2mSNU, so mTNU = 2(28) or 56.
Example 4

33. mUPR + mRMT + mTNU = 180 Triangle Angle Sum Theorem
Use the Incenter Theorem
mUPR + mRMT + mTNU = 180 Triangle Angle Sum Theorem
mUPR = 180 Substitution
mUPR = 180 Simplify.
mUPR = 62 Subtract from each side.
Since PS bisects UPR, 2mSPU = mUPR. This means that mSPU = mUPR. __ 1 2
Answer: mSPU = (62) or 31 __ 1 2
Example 4

34. A. Find the measure of GF if D is the incenter of ΔACF.
B. 144
C. 8
D. 65
Example 4

35. B. Find the measure of BCD if D is the incenter of ΔACF.
Example 4