Lesson 10.3 Perpendiculars in Space pp. 423-427

1. Lesson 10.3 Perpendiculars in Space pp. 423-427

2. Objectives: 1. To define perpendicular figures in space. 2. To prove theorems involving perpendiculars in space.

3. Definition Perpendicular planes are two planes that form right dihedral angles.

4. D B C E A D-AB-E is a right dihedral angle.

5. Definition A line perpendicular to a plane is a line that intersects the plane and is perpendicular to every line in the plane that passes through the point of intersection.

7. Definition A perpendicular bisecting plane of a segment is a plane that bisects a segment and is perpendicular to the line containing the segment.

8. x x

9. Theorem 10.2 A line perpendicular to two intersecting lines in a plane is perpendicular to the plane containing them.

10. l m p A n

11. Theorem 10.3 If a plane contains a line perpendicular to another plane, then the planes are perpendicular.

12. n D A E B m C

13. Theorem 10.4 If intersecting planes are each perpendicular to a third plane, then the line of intersection of the first two is perpendicular to the third plane.

14. Theorem 10.5 If AB is perpendicular to plane p at B, and BC  BD in plane p, then AC  AD.

15. Theorem 10.5 A p B D C

16. Theorem 10.6 Every point in the perpendicular bisecting plane of segment AB is equidistant from A and B.

17. Theorem 10.7 The perpendicular is the shortest segment from a point to a plane.

18. Two planes perpendicular to the same plane are parallel? 1. True 2. False

19. Two lines perpendicular to the same plane are parallel? 1. True 2. False

20. Two planes perpendicular to the same line are parallel? 1. True 2. False

21. Homework p. 427

22. ►A. Exercises Make a sketch of the following. 1. A dihedral linear pair

23. A p B D C ►A. Exercises Make a sketch of the following. 3. Theorem 10.5

24. ►A. Exercises Make a sketch of the following. 5. Theorem 10.7

25. B • D • X • A C ►A. Exercises Use the diagram for exercises 6-7. 7. State an Angle Addition Theorem for Dihedral Angles.

26. ►A. Exercises Explain or define each term below. 9. Complementary dihedral angles

27. ►B. Exercises Draw conclusions about dihedral angles (based on your knowledge of plane angles). 11. Vertical dihedral angles are . . .

28. ►B. Exercises Draw conclusions about dihedral angles (based on your knowledge of plane angles). 13. Supplementary dihedral angles that are adjacent . . .

29. ►B. Exercises Draw conclusions about dihedral angles (based on your knowledge of plane angles). 15. If one dihedral angle of a dihedral linear pair is a right angle, then . . .

30. Given: m ⊥ p, n ⊥ p, m intersects n Prove: AB ⊥ p m A n p C E B F D ►C. Exercises Prove each theorem below. 16. Theorem 10.4

31. StatementsReasons 1. m  p, n  p, & 1. Given m intersects n 2. m  n = AB 2. Plane Intersection Postulate 3. A-BC-D and 3. Definition of A-BE-F are perpendicular right angles planes

32. StatementsReasons 4. mA-BC-D=90 4. Def. of right mA-BE-F=90 dihedral angles 5. mA-BC-D= 5. Def. of dihedral mABD angle measure mA-BE-F= mABF 6. mABD=90 6. Transitive prop. mABF=90 of equality

33. StatementsReasons 7. ABD & ABF 7. Def. of right are right angles angle 8. AB  DB 8. Def. of perpend. AB  BF lines 9. AB  p 9. Line perpend. to two intersecting lines is perpend. to the plane containing them

34. A Given: AB ⊥ p at B, BC ≈ BD C, D  p Prove: AC  AD p B D C ►C. Exercises Prove each theorem below. 17. Theorem 10.5

35. ■ Cumulative Review 19. Name two postulates for proving triangles congruent.

36. ■ Cumulative Review 20. Name two theorems for proving triangles congruent.

37. ■ Cumulative Review 21. Name a fifth method for proving triangles congruent that works only for right triangles.

38. ■ Cumulative Review 22. What is SAS called when applied to right triangles?

39. ■ Cumulative Review 23. What is ASA called when applied to right triangles?