138. Polygon. A portion of a plane bounded by a broken line is called a polygon. The terms sides, perimeter, angles, vertices, and diagonals are employed in the usual sense in connection with polygons in general. 139. Polygons classified as to Sides. A polygon is a triangle, if it has three sides; a quadrilateral, if it has four sides; a hexagon, if it has six sides. These names are sufficient for most cases. The next few names in order are heptagon, octagon, nonagon, decagon, undecagon, dodecagon. A polygon is equilateral, if all of its sides are equal. 140. Polygons classified as to Angles. A polygon is equiangular, if all of its angles are equal; convex, if each of its angles is less than a straight angle; concave, if it has an angle greater than a straight angle. Equilateral Equiangular Hexagon Convex Concave An angle of a polygon greater than a straight angle is called a reëntrant angle. When the term polygon is used, a convex polygon is understood. 141. Regular Polygon. A polygon that is both equiangular and equilateral is called a regular polygon. 142. Relation of Two Polygons. Two polygons are mutually equiangular, if the angles of the one are equal to the angles of the other respectively, taken in the same order; mutually equilateral, if the sides of the one are equal to the sides of the other respectively, taken in the same order; congruent, if mutually equiangular and mutually equilateral, since they then can be made to coincide. PROPOSITION XXXII. THEOREM 143. 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. F B Given the polygon ABCDEF, having n sides. = (n − 2) 2 rt. 4. Proof. From A draw the diagonals AC, AD, AE. The sum of the ▲ of the ▲ is equal to the sum of the of the polygon. Now there are (n − 2) ▲. Ax. 11 (For there is one ▲ for each side except the two sides adjacent to A.) The sum of the of each ▲ = 2 rt. . § 107 .. the sum of the ▲ of the (n − 2) ▲, that is, the sum of the of the polygon, is equal to (n-2)2 rt. 4, by Ax. 3. Q. E. D. 144. COROLLARY 1. The sum of the angles of a quadrilateral equals four right angles; and if the angles are all equal, each is a right angle. 145. COROLLARY 2. Each angle of a regular polygon of n 2(n-2) sides is equal to n right angles. EXERCISE 12 1. What is the sum of the angles of (a) a pentagon? (b) a hexagon? (c) a heptagon? (d) an octagon? (e) a decagon? (f) a dodecagon? (g) a polygon of 24 sides? 2. What is the size of each angle of (a) a regular pentagon ? (b) a regular hexagon? (c) a regular octagon? (d) a regular decagon? (e) a regular polygon of 32 sides? 3. How many sides has a regular polygon, each angle of which is 1 right angles? 4. How many sides has a regular polygon, each angle of which is 13 right angles? 5. How many sides has a regular polygon, each angle of which is 108° ? 6. How many sides has a regular polygon, each angle of which is 140° ? 7. How many sides has a regular polygon, each angle of which is 156° ? 8. Four of the angles of a pentagon are 120°, 80°, 90°, and 100° respectively. Find the fifth angle. 9. Five of the angles of a hexagon are 100°, 120°, 130′, 150°, and 90° respectively. Find the sixth angle. 10. The angles of a quadrilateral are x, 2x, 2x, and 3x. How many degrees are there in each? 11. The angles of a quadrilateral are so related that the second is twice the first, the third three times the first, and the fourth four times the first. How many degrees in each? 12. The angles of a hexagon are x, 21⁄2 x, 31⁄2 x, 2 x, 2x, and x. How many degrees are there in each? 13. The sum of two angles of a triangle is 100° and their difference is 40°. How many degrees are there in each of the three angles of the triangle? PROPOSITION XXXIII. THEOREM 146. The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is equal to four right angles. Given the polygon ABCDE, having its n sides produced in succession. To prove that the sum of the exterior = 4rt. . Proof. Denote the interior of the polygon by a, b, c, d, e, and the corresponding exterior by a', b', c', d', e'. Then, considering each pair of adjacent angles, and Za+Za'a st. 4, = 2b+2b'= a st. Z. (The two adjacent & which one straight line makes with another are together equal to a straight 2.) In like manner, each pair of adj. = a st.. But the polygon has n sides and ʼn angles. Therefore the sum of the interior and exterior to n st. 4, or 2 n rt. . § 43 is equal Ax. 3 ,. the sum of the exterior <= 4 rt. 4, by Ax. 2. Q.E.D. EXERCISE 13 1. An exterior angle of a triangle is 130° and one of the opposite interior angles is 52°. Find the number of degrees in each angle of the triangle. 2. Two consecutive angles of a rectangle are bisected by lines meeting at P. How many degrees in the angle P? 3. Two angles of an equilateral triangle are bisected by lines meeting at P. How many degrees in the angle P? 4. The two base angles of an isosceles triangle are bisected by lines meeting at P. The vertical angle of the triangle is 30°. How many degrees in the angle P? 5. The vertical angle of an isosceles triangle is 40°. This and one of the base angles are bisected by lines meeting at P. How many degrees in the angle P? 6. One exterior angle of a parallelogram is one eighth of the sum of the four exterior angles. How many degrees in each angle of the parallelogram? 7. How many degrees in each exterior angle of a regular hexagon? of a regular octagon ? 8. In a right triangle one acute angle is twice the other. How many degrees in each exterior angle of the triangle? 9. Make out a table showing the number of degrees in each interior angle and each exterior angle of regular polygons of three, four, five, ..., ten sides. 10. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 11. In this parallelogram ABCD, AP= CR, and BQ = DS. Prove that PQRS is S also a parallelogram. A R B 12. If the mid-points of the sides of a parallelogram are connected in order, the resulting figure is also a parallelogram. |