Some technical terms to know about Angle between two lines. B. Angle between a line and a plane. C. Intersection of two planes. D. Angle between two planes. E. A line of greatest slope.

A. Angle between two lines Whenever ambiguous, always refer to the acute angle.

B. Angle between a line and a plane A’ is the projection of the point A onto the plane  if AA’ is perpendicular to  A  A’

D. Angle between a plane and a line Line segment A’B’is the projection of A on the plane  B A  A’ B’

B. Angle between a line and a plane If the line AB intersects the line at A, then we say that B’ is the projection of B on  and AB is the line of projection B  A B’

B. A line and a plane(2) (a) If the line is parallel to the plane, we cannot find the point of intersection L 

B. A line and a plane(3) (b) A normal of a plane is a line perpendicular to every line lying on the plane L L1 L2 L3

B. A line and a plane(1) (c) If a line is not parallel to the plane, then we can find the point of intersection. L 

B. Angle between a line and a plane QN is a perpendicular to the plane ABCD. PN is the projection of PQ on the plane ABCD is the angle between the line PQ and the plane ABCD.

C. Angle between 2 planes(1) Eg. Two walls which are opposite to each other (a)Two parallel planes 1 2

C. Angle between two planes(2) (b) Line of intersection of the planes 1 E.g. 2

C. Angle between two planes Find the line of intersection. Construct two lines which lie on the planes such that both of them are perpendicular to the line of intersection Can you now define the angle between two planes?

E. Line of greatest slope If a horizontal plane which intersects an inclined plane ( ) at AB and L is a line on , L is the line of greatest slope on  if L is perpendicular to AB.  L A B

E. Line of greatest slope In the figure , XYAB and XYCD are two intersecting planes which XYCD is inclined and XYAB is horizontal , PQ is a line on XYCD and perpendicular to XY。 Notice that  >  The angle between the line of greatest slope and the horizontal plane is also the angle between the planes

E. Line of greatest slope D C X Y B P Q R   A

A Find the angle between OB and ABCD.   O Find the angle between OB and ABCD. A B D N What is the projection of OB on the plane ABCD? What is the angle between OB and ABCD? C

Find the angle between ON and plane OBC M Find the angle between ON and plane OBC What is the projection of ON and plane OBC? What is the required angle? C B

Find the angle between planes OBC and ABCD What is the line of intersection? Draw a line on the plane OBC and ⊥BC Draw a line on the plane ABCD and ⊥ BC. C M What is the required angle?

Find the angle between planes ONB and ONM What is the line of intersection? Draw a line on ONB and ⊥ON C Draw a line on ONM and ⊥ON B What is the required angle?

Angle between planes OAB and OBC M What is the line of intersection? Draw a line on OAB and⊥OB Draw a line on OBC and ⊥OB C What is the required angle? B

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