lesson 10.3 perpendiculars in space pp. 423-427
DESCRIPTION
Lesson 10.3 Perpendiculars in Space pp. 423-427. Objectives: 1.To define perpendicular figures in space. 2.To prove theorems involving perpendiculars in space. Definition. Perpendicular planes are two planes that form right dihedral angles. D. B. C. E. A. - PowerPoint PPT PresentationTRANSCRIPT
Lesson 10.3Perpendiculars in Space
pp. 423-427
Lesson 10.3Perpendiculars in Space
pp. 423-427
Objectives:1. To define perpendicular figures in
space.2. To prove theorems involving
perpendiculars in space.
Objectives:1. To define perpendicular figures in
space.2. To prove theorems involving
perpendiculars in space.
Perpendicular planesPerpendicular planes are two are two planes that form right dihedral planes that form right dihedral angles.angles.
DefinitionDefinitionDefinitionDefinition
AA
BBDD
CCEE
D-AB-E is a right dihedral angle.D-AB-E is a right dihedral angle.
A A line perpendicular to a planeline perpendicular to a plane is a line that intersects the is a line that intersects the plane and is perpendicular to plane and is perpendicular to every line in the plane that every line in the plane that passes through the point of passes through the point of intersection.intersection.
DefinitionDefinitionDefinitionDefinition
A A perpendicular bisecting perpendicular bisecting plane plane of a segment is a plane of a segment is a plane that bisects a segment and is that bisects a segment and is perpendicular to the line perpendicular to the line containing the segment.containing the segment.
DefinitionDefinitionDefinitionDefinition
x x
Theorem 10.2A line perpendicular to two intersecting lines in a plane is perpendicular to the plane containing them.
Theorem 10.2A line perpendicular to two intersecting lines in a plane is perpendicular to the plane containing them.
AA
ll
mm
nn
pp
Theorem 10.3If a plane contains a line perpendicular to another plane, then the planes are perpendicular.
Theorem 10.3If a plane contains a line perpendicular to another plane, then the planes are perpendicular.
mm
nn
CC
DD
BBEE
AA
Theorem 10.4If intersecting planes are each perpendicular to a third plane, then the line of intersection of the first two is perpendicular to the third plane.
Theorem 10.4If intersecting planes are each perpendicular to a third plane, then the line of intersection of the first two is perpendicular to the third plane.
Theorem 10.5If AB is perpendicular to plane p at B, and BC BD in plane p, then AC AD.
Theorem 10.5If AB is perpendicular to plane p at B, and BC BD in plane p, then AC AD.
Theorem 10.5Theorem 10.5
pp
AA
BB
CCDD
Theorem 10.6Every point in the perpendicular bisecting plane of segment AB is equidistant from A and B.
Theorem 10.6Every point in the perpendicular bisecting plane of segment AB is equidistant from A and B.
Theorem 10.7The perpendicular is the shortest segment from a point to a plane.
Theorem 10.7The perpendicular is the shortest segment from a point to a plane.
Two planes perpendicular to the same plane are parallel?
1. True 2. False
Two planes perpendicular to the same plane are parallel?
1. True 2. False
Two lines perpendicular to the same plane are parallel?
1. True 2. False
Two lines perpendicular to the same plane are parallel?
1. True 2. False
Two planes perpendicular to the same line are parallel?
1. True 2. False
Two planes perpendicular to the same line are parallel?
1. True 2. False
Homeworkp. 427
Homeworkp. 427
►A. ExercisesMake a sketch of the following.
1. A dihedral linear pair
►A. ExercisesMake a sketch of the following.
1. A dihedral linear pair
►A. ExercisesMake a sketch of the following.
3. Theorem 10.5
►A. ExercisesMake a sketch of the following.
3. Theorem 10.5
pp
AA
BB
CCDD
►A. ExercisesMake a sketch of the following.
5. Theorem 10.7
►A. ExercisesMake a sketch of the following.
5. Theorem 10.7
►A. ExercisesUse the diagram for exercises 6-7.
7. State an Angle Addition Theorem for Dihedral Angles.
►A. ExercisesUse the diagram for exercises 6-7.
7. State an Angle Addition Theorem for Dihedral Angles.
•D•D•
X•X•
A•A
BB
CC
►A. ExercisesExplain or define each term below.
9. Complementary dihedral angles
►A. ExercisesExplain or define each term below.
9. Complementary dihedral angles
►B. ExercisesDraw conclusions about dihedral angles (based on your knowledge of plane angles).
11. Vertical dihedral angles are . . .
►B. ExercisesDraw conclusions about dihedral angles (based on your knowledge of plane angles).
11. Vertical dihedral angles are . . .
►B. ExercisesDraw conclusions about dihedral angles (based on your knowledge of plane angles).
13. Supplementary dihedral angles that are adjacent . . .
►B. ExercisesDraw conclusions about dihedral angles (based on your knowledge of plane angles).
13. Supplementary dihedral angles that are adjacent . . .
►B. ExercisesDraw 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 . . .
►B. ExercisesDraw 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 . . .
►C. ExercisesProve each theorem below.
16. Theorem 10.4
►C. ExercisesProve each theorem below.
16. Theorem 10.4
Given: m ⊥ p, n ⊥ p,
m intersects nProve: AB ⊥ p
Given: m ⊥ p, n ⊥ p,
m intersects nProve: AB ⊥ p
EE
AA
CC
nnpp
mm
BB
DDFF
Statements ReasonsStatements Reasons1. m p, n p, & 1. Given
m intersects n2. m n = AB 2. Plane Intersection
Postulate3. A-BC-D and 3. Definition of
A-BE-F are perpendicularright angles planes
1. m p, n p, & 1. Givenm intersects n
2. m n = AB 2. Plane IntersectionPostulate
3. A-BC-D and 3. Definition of A-BE-F are perpendicularright angles planes
Statements ReasonsStatements Reasons4. mA-BC-D=90 4. Def. of right
mA-BE-F=90 dihedral angles5. mA-BC-D= 5. Def. of dihedral
mABD angle measuremA-BE-F=mABF
6. mABD=90 6. Transitive prop.mABF=90 of equality
4. mA-BC-D=90 4. Def. of rightmA-BE-F=90 dihedral angles
5. mA-BC-D= 5. Def. of dihedralmABD angle measuremA-BE-F=mABF
6. mABD=90 6. Transitive prop.mABF=90 of equality
Statements ReasonsStatements Reasons7. ABD & ABF 7. Def. of right
are right angles angle8. AB DB 8. Def. of perpend.
AB BF lines9. AB p 9. Line perpend. to
two intersecting lines is perpend. to the plane containing them
7. ABD & ABF 7. Def. of rightare 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
►C. ExercisesProve each theorem below.
17. Theorem 10.5
►C. ExercisesProve each theorem below.
17. Theorem 10.5
AA
pp
BBCC
DD
Given: AB ⊥ p at B,BC ≈ BDC, D p
Prove: AC AD
Given: AB ⊥ p at B,BC ≈ BDC, D p
Prove: AC AD
■ Cumulative Review19. Name two postulates for
proving triangles congruent.
■ Cumulative Review19. Name two postulates for
proving triangles congruent.
■ Cumulative Review20. Name two theorems for
proving triangles congruent.
■ Cumulative Review20. Name two theorems for
proving triangles congruent.
■ Cumulative Review21. Name a fifth method for
proving triangles congruent that works only for right triangles.
■ Cumulative Review21. Name a fifth method for
proving triangles congruent that works only for right triangles.
■ Cumulative Review22. What is SAS called when
applied to right triangles?
■ Cumulative Review22. What is SAS called when
applied to right triangles?
■ Cumulative Review23. What is ASA called when
applied to right triangles?
■ Cumulative Review23. What is ASA called when
applied to right triangles?