show that ij coincides with y? Write your proof and state theorem.

Call it Prop. V.

Can you prove this theorem by projecting one line?

Z|| plane x P. [?]

Z || y [?]

y lies in plane x P. [?]

x|line JP I. [?]

y lies in plane x P and passes through P;

... JPI and y must coincide. [?] Q. E. D.

EXERCISES.

378. Can you show that any theorem in Plane Geometry in regard to a A is true also in Solid Geometry?

Illustrate your exercises with good drawings.

379. If a plane is passed through each of 2 || lines, the intersection of the planes is || to each of the lines.

380. If 3 planes meet in 3 lines, the intersections either meet in a point or they are ||.

381. If each of the two intersecting lines is || to a plane, the plane of these lines is || to the first plane.

382. Parallel lines included between || planes are equal. 383. Construct through a given point a plane || to a given plane.

384. Construct through a given line a plane || to a given line.

385. Construct through a given point a plane || to each 2 lines. Is this ever impossible? Discuss.

388.

PROPOSITION VI.

Stand a book on the desk partly open and suppose the pages to represent parallelograms. Compare the angle made by the two top edges with the angle made by the two corresponding lower edges. Can you illustrate the same truth in the room?

In the figure suppose a and b two intersecting lines and a' and 'two intersecting lines respectively || to a and b.

Compare their Zs.

Call the intersections O and O'. Join these points. From any points on a and b, as P and Q, erect ||s to O O'.

(1) Can you pass a plane through a, a', and O O'? (2) Will the parallel from P meet a'? Why? Letter the point, P'.

(3) Will parallel through Q intersect 6? Letter intersection Q?

(4) Compare O P and O' P'; O Q and O' Q'.

(5) Join P Q and P' Q'. How are P Q and P'Q' related? (6) Compare / POQ with Z P'O' Q'.

(7) Show that

formed by a' and b'.

POQ is supplementary to two angles

Draw conclusion, and call it Prop. VI.

389.

Prove that if each of two intersecting lines is || to a plane, the plane of these lines is || to the first plane.

Use reductio ad absurdum. tersecting lines meets the given

Suppose the plane of the inplane in line x. Then each

of the intersecting lines meets the plane. (Pupil finish.) Call this Prop. VII.

EXERCISES.

386. If a line cuts one of two || lines, must it cut the other? Are the corresponding s equal.

387.

equal.

Prove that || lines included between || planes are

388. If 2 || lines intersect a plane, compare the angles formed.

389. If a straight line intersects 2 || planes, compare the angles formed.

390. If a line is || to each of 2 intersecting planes, how is it related to their intersection?

DEFINITIONS.

390.

The distance from a point to a plane is the perpendicular distance.

391.

The point where a 1 meets a plane is called the foot of the perpendicular.

392.

A line is said to be 1, or normal to a plane, when it is 1 to every line in that plane which passes through its foot. When is a line oblique to a plane?

393.

Lines or points which lie in the same plane are coplanar

394.

(1) Three or more points which lie in the same line are said to collinear.

(2) A line is || to a plane if it cannot meet it however far produced. The plane is said to be || to the line.

395.

The projection of a point on a plane is the foot of the 1 from the point to the plane. Illustrate with your pencil and desk.

396.

The projection of a line on a plane is a straight line joining the projections of the extremities of the line on the plane. In the figure, a, b, c, d are projections of the points A, B, C, D and line a d in the projection of line A D on the plane M N.

The smaller angle formed by a line and its projection is called the inclination of the line to the plane.

398.

The angle which a line makes with a plane is the angle which it makes with its projection. Illustrate with pencil and

desk.

399.

The plane which a line makes with its projection is called the projecting plane.

400.

Where are all the common points of 2 planes?

391.

EXERCISES.

How many planes are determined by 6 points, 3 being collinear?

392. How many planes in general are determined by 4 points in space, no 3 being collinear?

393. A point, P, is in three planes, P, Q, R. Is it necessarily fixed?

In the figure suppose line a 1 to both b and d. How is a related to auy other line, as c, lying in the plane of b and d and passing through their intersection?

Sug. I. On line a make O P = O P'; draw any line cutting b, c, d in E, F, G, and join points with P and P'.