342. Show that a regular dodecaedron may be inscribed in a regular icosaedron.

343. Show that a regular icosaedron may be inscribed in a regular dodecaedron.

344. Find the radii of the sphere inscribed in a regular tetraedron whose edge is 12 ft.

345. To find a method of bisecting a given arc or given angle of a sphere.

346. There is a cube inscribed in a sphere; the surface of the cube is equal to the surface of the sphere. Find the diameter of the sphere. Are these conditions possible?

347. The angle which a line makes with a plane is the angle between the line and its projection upon the plane. Prove that if a line intersect two parallel planes it makes equal angles with them.

348. Prove that only one common perpendicular can be drawn to two lines not in the same plane.

THEOREMS OF PLANE GEOMETRY.

349. The demonstrations of the preceding pages refer to a few of the theorems of Plane Geometry. These theorems are collected here and numbered the same as in the text; they will be found convenient for reference.

350. At a point in a straight line only one perpendicular to that line can be drawn; and from a point without a straight line only one perpendicular to that line can be drawn.

351. Two parallel lines cannot meet.

352. Two lines perpendicular to the same straight line are parallel to each other.

353.

Any side of a triangle is less than the sum of the other two sides.

354. The sum of the three angles of a triangle is equal to two right angles.

355. Two triangles are equal in all respects when two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.

356. Two triangles are equal when a side and two adjacent angles of the one are respectively equal to a side and two adjacent angles of the other.

357. Two triangles are equal when the three sides of the one are equal respectively to the three sides of the other.

358. If two sides of a triangle be equal respectively to two sides of another, but the third side of the first triangle be greater than the third side of the second, then the angle opposite the third side of the first triangle is greater than the angle opposite the third side of the second.

359. The diagonal of a parallelogram divides the figure into two equal triangles.

360. In a parallelogram the opposite sides are equal and the opposite angles are equal.

361. If a line be drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.

362. Two triangles which have their sides respectively parallel are similar.

363. Parallelograms having equal bases and equal altitudes are equivalent.

364. The area of a triangle is equal to one-half the product of its base by its altitude.

365. Triangles having equal bases are to each other as their altitudes; triangles having equal altitudes are to each

other as their bases; any two triangles are to each other as the product of their bases by their altitudes.

366. The area of a trapezoid is equal to one-half the sum of the parallel sides multiplied by the altitude.

367. I. A circle may be circumscribed about a regular polygon.

II. A circle may be inscribed in a regular polygon. 368. If a line is perpendicular to one of two parallel lines it is perpendicular to the other also.

369. Parallel Lines are straight lines which lie in the same plane and have the same direction, or opposite directions.

370. Similar Polygons are polygons which have their homologous angles equal and their homologous sides proportional.

371. A Parallelogram is a quadrilateral which has its opposite sides parallel.

372. The area of a parallelogram is equal to the product of its base and altitude.

373. The area of a circle equals the square of the radius multiplied by л, i.e. An R2.

π

374. In the same or equal circles equal chords subtend equal arcs.

375. The angle at the centre of a circle is measured by the arc of the circumference intercepted by the sides of the angle.

376. A tangent to a circle is perpendicular to a radius drawn to the point of tangency.

377. The line which joins the middle points of the two sides, not parallel, of a trapezoid, equals half the sum of the parallel sides.