Lesson 10.3 Perpendiculars in Space pp. 423-427

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?