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Splash Screen. 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

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  1. Splash Screen

  2. 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

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

  4. Find x if measure angle A = 10x + 15, measure angle B = 8x – 18, and measure angle C = 12x + 3. A. 3.75 B. 6 C. 12 D. 16.5 5-Minute Check 2

  5. A. B. C. D. , Name the corresponding congruent sides if ΔLMNis congruent to ΔOPQ. 5-Minute Check 4

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

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

  8. perpendicular bisector • concurrent lines • point of concurrency • circumcenter • incenter Vocabulary

  9. perpendicular bisector • a segment which bisects a line segment into two equal parts at 90° • concurrent lines • point of concurrency • circumcenter • incenter Vocabulary

  10. perpendicular bisector • concurrent lines • three or more lines in a plane or higher-dimensional space that intersect at a single point. • point of concurrency • circumcenter • incenter Vocabulary

  11. perpendicular bisector • concurrent lines • point of concurrency • the point where three or more lines intersect • circumcenter • incenter Vocabulary

  12. perpendicular bisector • concurrent lines • point of concurrency • circumcenter • the point where the perpendicular bisectors of the sides intersect. • incenter Vocabulary

  13. perpendicular bisector • concurrent lines • point of concurrency • Circumcenter • incenter • a triangle’s center Vocabulary

  14. Concept

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

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

  17. 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

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

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

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

  21. Concept

  22. Concept

  23. 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

  24. 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

  25. 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

  26. Concept

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

  28. Use the Angle Bisector Theorems B. Find measure angle WYZ. Example 3

  29. Use the Angle Bisector Theorems WYZ≅XYW Definition of angle bisector m WYZ = m XYW Definition of congruence WYZ = 28 Substitution Answer:mWYZ = 28 Example 3

  30. 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

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

  32. B. Find the measure of HFI. A. 28 B. 30 C. 15 D. 30 Example 3

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

  34. Concept

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

  36. 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, ST = 6. Answer:ST = 6 Example 4

  37. Since MS bisects RMT, m RMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, m TNU = 2mSNU, so mTNU = 2(28) or 56. Use the Incenter Theorem B. Find m SPU if S is the incenter of ΔMNP. Example 4

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

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

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

  41. Splash Screen

  42. In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11. A. –5 B. 0.5 C. 5 D. 10 5-Minute Check 1

  43. In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13. A. 13 B. 11 C. 7 D. –13 5-Minute Check 2

  44. In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11). A. –12.5 B. 2.5 C. 10.25 D. 12.5 5-Minute Check 3

  45. In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ ___ A.DE B.DA C.DC D.DB ___ ___ ___ 5-Minute Check 4

  46. In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF? A.GCD B.DCG C.DFB D.ADE 5-Minute Check 5

  47. Which of the following statements about the circumcenter of a triangle is false? A. It is equidistant from the sides of the triangle. B. It can be located outside of the triangle. C. It is the point where the perpendicular bisectors intersect. D. It is the center of the circumscribed circle. 5-Minute Check 6

  48. You identified and used perpendicular and angle bisectors in triangles. • Identify and use medians in triangles. • Identify and use altitudes in triangles. Then/Now

  49. median • centroid • altitude • orthocenter Vocabulary

  50. Median • a line segment joining a vertex to the midpoint of the opposing side • centroid • altitude • orthocenter Vocabulary

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