Wentworth's Plane and Solid Geometry · Book VI, Exs. 616–629 · Book 7: Book VI: Lines and Planes in Space
92. Exercises on lines and planes
Goal: Your student works several loci and theorems about lines, planes and trihedral angles.
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
Page 288: theorems and loci (Ex. 616–629) closing Book VI.
The lesson, step by step
Warm up
Ask your student to name the four main topics of Book VI. (Lines and planes, dihedral angles, polyhedral angles, projections.)
Loci
Do these together with cards and pencils:
Ex. 617. Find the locus of points in space equidistant from all points in the circumference of a circle.
Ex. 618. Find the locus of points in a plane equidistant from a given point without the plane.
Ex. 619. Find a point at equal distances from four points not all in the same plane.
Theorems
Your student chooses two:
Ex. 621. The projections on a plane of equal and parallel lines are equal and parallel.
Ex. 622. If two face angles of a trihedral angle are equal, the dihedral angles opposite them are equal.
or Ex. 620, 623 or 625.
Review the Book
Go back over Props. I–XXVIII. Your student states each one aloud and makes a quick sketch, as the author advises at the end of each Book.
Wrap up
Your student reads one proof aloud.
Answers to the book's problems
Ex. 616: two planes, each perpendicular to the plane of the two lines, passing through the bisectors of the angles the lines make. Ex. 617: the straight line through the centre perpendicular to the plane of the circle (§ 516). Ex. 618: a circle in the plane whose centre is the foot of the perpendicular from the point. Ex. 619: the planes perpendicular to three of the lines joining the points at their middle points meet in one point (as for the circumscribed sphere, Book VIII, Prop. VII). Ex. 621: the projecting planes are parallel; use § 529. Ex. 622: an isosceles trihedral angle can be laid on its symmetrical angle with the equal faces changing places (Ex. 625), so the two dihedral angles fall on each other. Ex. 626: a straight line through the vertex, where the planes of points equidistant from each pair of edges meet (compare Ex. 616). Ex. 627: the straight line in which the three bisecting planes meet (Ex. 623).
Worked out for this site from the scan. If a number in your copy differs, trust the book.