« ΠροηγούμενηΣυνέχεια »
ON POLYGONS IN GENERAL. 143. Def. A Polygon is a plane figure bounded by straight lines.
144. DEF. The bounding lines are the sides of the polygon, and their sum, as A B + B C + C D, etc., is the Perimeter of the polygon.
The angles which the adjacent sides make with each other are the angles of the polygon.
145. Def. A Diagonal of a polygon is a line joining the vertices of two angles not adjacent. B
146. DEF. An Equilateral polygon is one which has all its sides equal.
147. DEF. An Equiangular polygon is one which has all its angles equal.
148. DEF. A Convex polygon is one of which no side, when produced, will enter the surface bounded by the perimeter.
149. Def. Each angle of such a polygon is called a Salient angle, and is less than two right angles.
150. DEF. A Concave polygon is one of which two or more sides, when produced, will enter the surface bounded by the perimeter.
151. DEF. The angle F D E is called a Re-entrant angle. When the term polygon is used, a convex polygon is nieant.
The number of sides of a polygon is evidently equal to the number of its angles.
By drawing diagonals from any vertex of a polygon, the fig. ure may be divided into as many triangles as it has sides less two. 152. DEF. Two polygons are Equal, when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed ; for the polygons can be applied to each other, and the corresponding triangles will evidently coincide. Therefore the polygons will coincide, and be equal in all respects.
153. Def. Two polygons are Mutually Equiangular, if the angles of the one be equal to the angles of the other, each to each, when taken in the same order; as the polygons ABCDEF, and A' B' C' D' E' F', in which ZA= LA', ZB= 2 B', ZC=2 C', etc.
154. DEF. The equal angles in mutually equiangular polygons are called Homologous angles; and the sides which lie between equal angles are called Homologous sides.
155. DEF. Two polygons are Mutually Equilateral, if the sides of the one be equal to the sides of the other, each to each, when taken in the same order.
Fig. 3. Two polygons may be mutually equiangular without being mutually equilateral; as Figs. 1 and 2.
And, except in the case of triangles, two polygons may be mutually equilateral without being mutually equiangular; as Figs. 3 and 4.
If two polygons be mutually equilateral and equiangular, they are equal, for they may be applied the one to the other so as to coincide.
156. DEF. A polygon of three sides is a Trigon or Triangle ; one of four sides is a Tetragon or Quadrilateral ; one of five sides is a Pentagon ; one of six sides is a Hexagon ; one of seven sides is a Heptagon; one of eight sides is an Octagon ; one of ten sides is a Decagon ; one of twelve sides is a Dodecagon.
PROPOSITION XLVI. THEOREM. 157. The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.
Let the figure ABCDEF be a polygon having n sides.
We are to prove
LA+ZB+ 2 C, etc., = 2 rt. $ (n − 2).
The sum of the 6 of the A = the sum of the angles of the polygon.
Now there are (n — 2) A, and the sum of the of each A = 2 rt. £. $ 98 .. the sum of the es of the A, that is, the sum of the of the polygon = 2 rt. És (n − 2).
Q. E. D. 158. COROLLARY. The sum of the angles of a quadrilateral equals two right angles taken (4 — 2) times, i. e. equals 4 right angles; and if the angles be all equal, each angle is a right angle. In general, each angle of an equiangular polygon of n sides is equal to 2 (n — 2) ,;
~ right angles.
PROPOSITION XLVII. THEOREM.
159. The exterior angles of a polygon, made by producing each of its sides in succession, are together equal to four right angles.
Let the figure A BCDE be a polygon, having its sides
produced in succession.
LA + La= 2 rt. 4,
(being sup.-adj. € ). ZB+2b = 2 rt. 28.
$ 34 In like manner each pair of adj. 6 = 2 rt. Es ;
.. the sum of the interior and exterior 1 = 2 rt. A taken as many times as the figure has sides, or,
2 n rt. 6. But the interior é = 2 rt. 6 taken as many times as the figure has sides less two, = 2 rt. (— 2), or,
2 n rt. 6 – 4 rt. .
Q. E. D.
1. Show that the sum of the interior angles of a hexagon is equal to eight right angles.
2. Show that each angle of an equiangular pentagon is fe of a right angle.
3. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles ?
4. How many sides has the polygon the sum of whose interior angles is equal to the sum of its exterior angles ?
5. How many sides has the polygon the sum of whose interior angles is double that of its exterior angles ?
6. How many sides has the polygon the sum of whose exterior angles is double that of its interior angles ?
7. Every point in the bisector of an angle is equally distant from the sides of the angle; and every point not in the bisector, but within the angle, is unequally distant from the sides of the angle.
8. BAC is a triangle having the angle B double the angle A. If B D bisect the angle B, and meet AC in D, show that BD is equal to A D.
9. If a straight line drawn parallel to the base of a triangle bisect one of the sides, show that it bisects the other also ; and that the portion of it intercepted between the two sides is equal to one half the base.
10. A B C D is a parallelogram, E and F the middle points of A D and B C respectively; show that B E and DF will trisect the diagonal A C.
11. If from any point in the base of an isosceles triangle parallels to the equal sides be drawn, show that a parallelogram is formed whose perimeter is equal to the sum of the equal sides of the triangle.
12. If from the diagonal B D of a square A B C D, B E be cut off equal to BC, and E F be drawn perpendicular to BD, show that D E is equal to E F, and also to FC.
13. Show that the three lines drawn from the vertices of a triangle to the middle points of the opposite sides meet in a