Wentworth's Plane and Solid Geometry · Book VI, Props. V–VII · Book 7: Book VI: Lines and Planes in Space
82. Oblique lines, loci in space, and parallel perpendiculars
Goal: Your student can compare oblique lines drawn from a point to a plane, name two loci in space, and prove that two lines perpendicular to the same plane are parallel.
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 V–VII (§§ 514–521) with their corollaries, including the loci of § 516 and § 517.
The lesson, step by step
Warm up
Ask: What is the distance from a point to a plane? Why is the perpendicular the shortest line?
The new idea
Two straight lines perpendicular to the same plane are parallel.
The locus of a point in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point.
Prop. V (§ 514): oblique lines meeting the plane at equal distances from the foot of the perpendicular are equal; the more remote one is the greater.
Prop. VI (§ 518) is the "theorem of three perpendiculars": from the foot of a perpendicular, draw a line at right angles to a line of the plane; the line from that point up to the perpendicular is also at right angles to it.
Draw the figure
Hold a pencil upright on the table. Tie a string to its top and swing the other end around on the table at a fixed length: it traces a circle about the foot (§ 515).
For Prop. VII, stand two pencils upright on the table. They look parallel; the proof shows why.
Walk through the proof
V: the right triangles ABC and ABD have AB common and BC = BD, so they are equal (§ 144). For unequal distances use § 101.
VI: take EC = ED; then BC = BD, so AC = AD (§ 514), and AE is perpendicular to CD (§ 161).
VII: draw AD and BD, and EF in the plane perpendicular to BD at D. Then CD, AD and BD are all perpendicular to EF, so they lie in one plane (§ 508). AB lies in that plane too, and AB and CD are both perpendicular to BD, so they are parallel (§ 104).
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
Ask: What is the locus of points equally distant from all points of a circle? (§ 516: the line through the centre perpendicular to its plane.) Show it with a pencil standing in the middle of a drawn circle.
Wrap up
Ask: If two lines are each parallel to a third line, must they be parallel to each other, even in space? (Yes, § 521.)