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Understanding Perpendicular and Angle Bisectors in Triangles

This lesson focuses on the key theorems related to perpendicular and angle bisectors in triangles. You will learn how to use the Perpendicular Bisector Theorem and Angle Bisector Theorem to find segment lengths and angles in various triangle types, including scalene, isosceles, and equilateral triangles. Real-world applications will be explored, alongside practice problems to solidify your understanding of these geometric concepts. Get ready for a five-minute review at the end to test your knowledge!

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Understanding Perpendicular and Angle Bisectors in Triangles

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Presentation Transcript

  1. Splash Screen

  2. Five-Minute Check (over Chapter 4) 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. A B C Classify the triangle. A. scalene B. isosceles C. equilateral 5-Minute Check 1

  4. A B C D 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

  5. A B C 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

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

  7. A B C D 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

  8. You used segment and angle bisectors. (Lesson 1–3 and 1–4) • Identify and use perpendicular bisectors in triangles. • Identify and use angle bisectors in triangles. Then/Now

  9. perpendicular bisector Vocabulary

  10. Concept

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

  12. Use the Perpendicular Bisector Theorems B. Find the measure of XY. Answer: 6 Example 1

  13. Use the Perpendicular Bisector Theorems C. Find the measure of 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

  14. A B C D A. Find the measure of NO. A. 4.6 B. 9.2 C. 18.4 D. 36.8 Example 1

  15. A B C D B. Find the measure of TU. A. 2 B. 4 C. 8 D. 16 Example 1

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

  17. Concept

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

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

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

  21. 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

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

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

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

  25. Splash Screen

  26. median • altitude Vocabulary

  27. Altitude:(of a triangle) is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Median:(of a triangle) is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side.

  28. Concept

  29. End of the Lesson

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