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Explore the significance of perpendicular bisectors, circumcenter, and incenter theorems in triangle geometry. Understand how to apply these theorems to solve real-world problems, ranging from finding congruent sides to determining angles. Delve into the concept of point of concurrency and learn about the circumcenter and incenter. Enhance your math skills by practicing with examples and checking your understanding with quick assessments. Develop a deeper understanding of triangle properties and geometric methods, enabling you to prove theorems and construct logical arguments effectively. Embrace the challenge of solving triangle-related problems and honing your mathematical practices.

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  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, andmC = 12x + 3. A. 3.75 B. 6 C. 12 D. 16.5 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. A. B. C. D. , Name the corresponding congruent sides if ΔLMN ΔOPQ. 5-Minute Check 4

  5. Find y if ΔDEF is an equilateral triangle and mF = 8y + 4. A. 22 B. 10.75 C. 7 D. 4.5 5-Minute Check 5

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

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

  12. 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. BILLIARDSA 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. 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. 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. 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. A. 28 B. 30 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. 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 = 100 82 = 64, 102 = 100 Example 4

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

  33. Since PS bisects UPR, 2mSPU = mUPR. This means that mSPU = mUPR. 1 1 __ __ 2 2 Answer:mSPU = (62) or 31 Use the Incenter Theorem mUPR + mRMT + mTNU = 180 Triangle Angle Sum Theorem mUPR + 62 + 56 = 180 Substitution mUPR + 118 = 180 Simplify. mUPR = 62 Subtract 118 from each side. Example 4

  34. A. Find the measure of GF if D is the incenter of ΔACF. A. 12 B. 144 C. 8 D. 65 Example 4

  35. B. Find the measure of BCD if D is the incenter of ΔACF. A. 58° B. 116° C. 52° D. 26° Example 4

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