Points, Lines and Planes in Solids

Angle between two straight lines In the figure, two non-parallel straight lines AB and CD intersect at a point E. The acute angle  formed is called the angle between two straight lines. D A E  B C

Distance between a point and a straight line The figure shows a point P lying outside the straight line AB. If a perpendicular line is drawn from P to meet AB at Q, then PQ is the distance between P and AB. P distance between point P and the straight line AB Q A B

Relationships between Lines and Planes On plane , L1 and L2 are two straight lines and they intersect at Q. P  L1 L2 Q If P is a point outside  such that PQ is perpendicular to both L1 and L2, then (i) PQ is perpendicular to plane , (ii) Q is the projection of P on plane , (iii) PQ is the distance between P and plane .

Consider a straight line AB intersects plane  at a point A. (i) AC is the projection of AB on plane , (ii) BAC is the angle between straight line AB and plane . B C  A If C is the projection of B on plane , then

Consider a line segment PQ that does not intersect plane , where S and T are the projections of P and Q on  respectively. P S Q R T  If PQ produced meets plane  at a point R, then (i) RS is the projection of RP on plane , (ii) TS is the projection of QP on plane , (iii) PRS is the angle between line segment PQ and plane .

Take cuboid ABCDHEFG below as an example: The projection of AG on plane ABCD is AC and the angle between AG and plane ABCD is GAC.

Follow-up question The figure shows a cuboid ABCDHEFG. (a) (i) Name the projection of CE on plane ABCD. (ii) Name the angle between CE and plane ABCD. (b) (i) Name the projection of CE on plane ADHE. (ii) Name the angle between CE and plane ADHE. Solution (a) (i) CA (ii) ECA (b) (i) DE (ii) CED

Relationships between Two Planes Two non-parallel planes will intersect at a straight line, and the straight line is called the line of intersection of the two planes. e.g. 1 and 2 intersect at a straight line AB and it is the line of intersection of the two planes. 1 2 A B

We can find the angle between two planes as follows: Q P (i) Locate a point P on the line of intersection AB of the two planes. (ii) On the plane ABEF, draw a straight line PQ that is perpendicular to AB.

QPR is the angle between planes ABEF and ABCD. (iii) On the plane ABCD, draw a straight line PR that is perpendicular to AB. QPR is the angle between planes ABEF and ABCD.

Consider the cuboid ABCDHEFG. The angle between plane ABCD and plane CDEF is FCB (or EDA). H G E F D C A B

Follow-up question The figure shows a cuboid ABCDHEFG. Name the angle between planes (a) HEBC and ABCD, (b) EBCH and BCGF. A B C D E H G F Solution (a) EBA or HCD (b) EBF or HCG