Wentworth's Plane and Solid Geometry · Book VI, Props. XIV–XV · Book 7: Book VI: Lines and Planes in Space
85. Angles with parallel sides; lines cut by parallel planes
Goal: Your student can prove that angles in space with parallel sides are equal and that parallel planes cut two lines proportionally.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)
What's in the book
Propositions XIV–XV (§§ 534–535) and Exercises 610–612.
The lesson, step by step
Warm up
Ask: State Prop. XIII. If two intersecting lines are each parallel to a plane, then ... ?
The new idea
If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal, and their planes are parallel.
If two straight lines are intersected by three parallel planes, their corresponding segments are proportional.
Draw the figure
For Prop. XV, draw three parallel planes as stacked parallelograms and two slanting lines AB and CD passing through them. Add the helping line AD.
Walk through the proof
XIV: take AD = A'D' and AC = A'C'. The figures ADD'A' and ACC'A' are parallelograms (§ 183), so CC' and DD' are equal and parallel, and CD = C'D'. The triangles ADC and A'D'C' are equal (§ 150), so the angles are equal.
XV: AD cuts the middle plane at G. In the plane of AB and AD, EG is parallel to BD; in the plane of AD and CD, GF is parallel to AC (§ 528). Then AE : EB = AG : GD = CF : FD (§ 342).
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Ex. 610. The line AB meets three parallel planes in the points A, E, B; and the line CD meets the same planes in the points C, F, D. If AE = 6 inches, BE = 8 inches, CD = 12 inches, compute CF and FD.
Then talk through Ex. 611 and 612: how would you draw a perpendicular to a plane, from a point outside it or at a point in it? (Use the method of Prop. IV.)
Wrap up
Ask: Where in a building do you see three parallel planes cut by slanting lines? (Floors cut by a staircase rail.)
Answers to the book's problems
Ex. 610: CF : FD = 6 : 8, and CF + FD = 12, so CF = 36/7 = 5 1/7 inches and FD = 48/7 = 6 6/7 inches. Ex. 611 and 612: follow the constructions in the proof of Prop. IV, Cases 2 and 1.
Worked out for this site from the scan. If a number in your copy differs, trust the book.