Theorem 21. The sum of the interior angles of an n-gon is (n-2) straight angles. Given To prove P, a polygon of n sides. that the sum of the interior angles is (n-2) straight angles. Proof. 1. P may be divided into (n-2) cross; for, (a) A 4-gon (quadrilateral) is a ▲+a▲, (b) A 5-gon (pentagon) is a 4-gon +a▲, (c) A 6-gon (hexagon) is a 5-gon +a▲, Th. 19 (d) And every addition of 1 side adds 1 A. (e) .. for an n-gon there are (n−2) A. 2. The sum of the of each ▲ is a st. Z. 3. .. the sum of the interior of an n-gon is (n − 2) st. ▲, ·.· these equal the sum of the of the A. COROLLARY. If each of two angles of a quadrilateral is a right angle, the other two angles are supplemental. (Why?) EXERCISES. 126. A general quadrilateral has three diagonals. Draw them. 127. How many diagonals in a common convex pentagon? hexagon? heptagon ? 128. How many points of intersection, at most, in a general quadrilateral? pentagon ? hexagon ? 129. How many diagonals, at most, has a general quadrilateral ? a general pentagon? a general hexagon ? 130. With the figure of th. 21, prove the theorem by connecting each vertex with a point O within the figure, thus forming n A, giving n st. 4, and then subtracting the two around O. 131. Investigate ex. 130 when O is on one of the sides and in such a position that the lines joining it to the vertices do not cut the sides. 132. Show that in a regular n-gon each angle equals 180° · (n − 2)/n. OPPOSITE SENSES OF LINE-SEGMENTS AND OF ANGLES. If a thermometer registers 70° above zero it is ordinarily stated, in scientific works, that it registers + 70°, while 10° below zero is indicated by — 10°, the sign changing from + to as the temperature decreases through zero. Similarly west longitude is represented by the sign +, while longitude on the other side of 0° (i.e. east) is represented by the sign the longitude changing its sign in passing through zero. A B C This custom is general in all sciences involving measurement, and hence in geometry. Thus in this figure, if the segment between B and C is thought of as extending from B to C, or generated by a point moving from B to C, it would be named BC; and, as is usually done in geometry with lines thought of as extending to the right, it would be considered a positive line. But if it be thought of as extending from C to B, it would be named CB, and considered a negative line. Then since BC moves the point from B to C, and CB moves it back to its original position, it is said that BC + CB = 0, an expression borrowed from algebra, where it would appear in a form like x + (−x) = 0. Similarly, AB BC + CA = 0, but AB + BC + AC = 2 AC. Similarly with regard to angles: the turning of an arm in a sense opposite to that of a clock-hand, counter-clockwise, is considered positive, while the turning in the opposite sense is considered negative. Thus, XOA is considered positive, but the acute AOX is considered negative, and this is indicated Z XOA acute AOX. case of lines, ≤ XOA + (− by the statement, XOA) = Z XOA + acute AOX – zero, and Z XOA + Z AOB+ acute A -X BOX = zero. This distinction in the sense of XOA and acute AOX is due to Möbius. Two magnitudes whose sum is zero, as AB and AB, are said to have the same absolute value. THE PRINCIPLE OF CONTINUITY. It will be noticed that th. 21 is true for both convex and concave polygons. Thus in the annexed figure it is true for the convex polygon P1ABC; if P1 moves to P2, it is still valid for polygon P P2ABC; if P2 moves to P3, it is still true if PABC is considered a quadrilateral with a straight angle at P3, and it is equally true if ABC is considered a triangle; if P3 passes to P P B P4, the theorem is still true for the concave polygon P4ABC; if P4 passes to P5, the figure PABC may be thought of as a quadrilateral in which BAP=0; finally, if P5 moves to P6, the quadrilateral becomes cross, ceases to have interior angles, in the ordinary +C sense of the word interior, yet even here the theorem is valid; by noticing that A has passed through zero and become negative, and that APC is the continuation of the interior angle at R1, P2, P3, P4, it is still true that A (negative) + ZB+ZC+ZP6 (reflex) = 2 st. angles. This leads to a principle of greatest importance in mathematics, which may thus be stated in an elementary way, but with sufficient completeness for the present work: +B PRINCIPLE OF CONTINUITY. Theorems proved for one figure continue true for general figures, so long as the given conditions continue, but zero and negative magnitudes may enter. Thus in the figure at the bottom of p. 50, the sum of the interior angles of P1ABC is a perigon; the figure may undergo changes, as of P1 moving to the right, and, however slight the movement, the condition still continues that the figure shall have interior angles; P may even pass through AB, and by attending to the signs of the angles, and properly extending the idea of interior angle, the conditions continue; the property is therefore continuous. The Principle of Continuity is largely due to the labors of Kepler, Boscovich, and Poncelet. EXERCISES. 133. Does the Principle of Continuity apply to th. 1? That is, taking B and B' as varying simultaneously, is the theorem valid if they are right angles? zero? if they pass through zero and become negative? if they become - 180° ? [In the cases of B=B' = 0, or 180°, or + 180°, or 360°, or any integral multiple of 180°, there are no triangles, but the figures are still congruent.] - 134. Similarly, consider such of the propositions 1-21 as the teacher directs. 15, BC = 135. A, B, C, D, E, are points on a line; AB 5, CD = 8, DE=-7. Draw the figure and compute AE. (In such numerical exercises, an eighth of an inch forms a convenient unit.) 136. Prove the statement made in the paragraph preceding the statement of the Principle of Continuity, that A (negative) + ZB+ZC +P (reflex) = 2 st. s. EXERCISES. 137. Each exterior angle of an eq how many times each interior angle? 138. Each exterior angle of a regular heptagon part of each interior angle? 139. Each exterior angle of a regular n-gon part of each interior angle? See if the result four 140. Is it possible for the exterior angle of a 70° ? 72°? 75°? 120°? 141. Prove th. 22 independently of th. 21 by t in the plane of the figure (inside or outside the p gon, or on the perimeter) and drawing parallels the sides from that point, and showing that the s of the exterior angles equals the perigon about t point. 142. If the student has proved ex. 141, let him 143. The quadrilateral formed by the bisector quadrilateral has its opposite angles supplemental. 144. Show that in ex. 143 the angles bisected interior or the four exterior angles. DEFINITIONS. A quadrilateral whose opposite sides are parallel is called a parallelogram. A quadrilateral that has one pair of opposite sides parallel is called a trapezoid. Trapezium is a term often applied to a quadrilateral no two of whose sides are parallel. By the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid. Parallel sides of a trapezoid are called its bases, and are distinguished as upper and lower. If the two opposite non-parallel sides of a trapezoid are equal, the trapezoid is said to be isosceles. In the above figures, angles A, B, or B, C, or C, D, or D, A are called consecutive angles. Angles A, C or B, D are called opposite angles. Theorem 23. Any two consecutive angles of a parallelogram are supplemental, and any two opposite angles are equal. COROLLARY. If one angle of a parallelogram is a right angle, all of its angles are right angles. (Why ?) DEFINITION. If one angle of a parallelogram is a right angle, the parallelogram is called a rectangle. By the corollary, all angles of a rectangle are right angles. |