나 2n-4)st. L'S. COROLLARY 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. ய Let ABCDE be a rectilineal figure of n sides. It is required to prove that all the interior angles +4_rt. 2° 2n rt. 4. Take any point O within the figure, and join O to each of its vertices. Then the figure is divided into n triangles. And the three of each together = 2 rt. 4°. But all the of all the ▲ make up all the interior angles of the figure together with the angles at O, which = 4 rt. ... all the int. 4s of the figure +4 rt. = 2n rt. 2o. Q.E.D. DEFINITION. A regular polygon is one which has all its sides equal and all its angles equal. Thus if D denotes the number of degrees in each angle of a regular polygon of n sides, the above result may be stated thus: nD +360° = n. 180°. EXAMPLE. Find the number of degrees in each angle of (i) a regular hexagon (6 sides); (ii) a regular octagon (8 sides); EXERCISES ON THEOREM 16. (Numerical and Graphical.) 1. ABC is a triangle in which the angles at B and C are respectively double and treble of the angle at A: find the number of degrees in each of these angles. 2. Express in degrees the angles of an isosceles triangle in which (i) Each base angle is double of the vertical angle; (i) Each base angle is four times the vertical angle. 3. The base of a triangle is produced both ways, and the exterior angles are found to be 94° and 126°; deduce the vertical angle. Construct such a triangle, and check your result by measurement. 4. The sum of the angles at the base of a triangle is 162°, and their difference is 60°: find all the angles. 5. The angles at the base of a triangle are 84° and 62°; deduce (i) the vertical angle (ii) the angle between the bisectors of the base angles. Check your results by construction and measurement. 6. In a triangle ABC, the angles at B and C are 74° and 62°; if AB and AC are produced, deduce the angle between the bisectors of the exterior angles. Check your result graphically. 7. Three angles of a quadrilateral are respectively 1141⁄2o, 50°, and 75°; find the fourth angle. 8. In a quadrilateral ABCD, the angles at B, C, and D are respectively equal to 2A, 3A, and 4A; find all the angles. 9. Four angles of an irregular pentagon (5 sides) are 40°, 78°, 122°, and 135°; find the fifth angle. 10. In any regular polygon of n sides, each angle contains right angles. 2(n-2) (i) Deduce this result from the Enunciation of Corollary 1. (ii) Prove it independently by joining one vertex A to each of the others (except the two immediately adjacent to A), thus dividing the polygon into n - 2 triangles. 11. How many sides have the regular polygons each of whose angles is (i) 108°, (ii) 156°? 12. Shew that the only regular figures which may be fitted together so as to form a plane surface are (i) equilateral triangles, (ii) squares, (iii) regular hexagons COROLLARY 2. If the sides of a rectilineal figure, which has no re-entrant angle, are produced in order, then all the exterior angles so formed are together equal to four right angles. く 1st Proof. Suppose, as before, that the figure has n sides; and consequently n vertices. Now at each vertex the interior + the exterior ≤ = 2 rt. 2o ; and there are n vertices, .. the sum of the int. + the sum of the ext. 4= 2n rt. ¿3. Take any point O, and suppose Oa, Ob, Oc, Od, and Oe, are lines parallel to the sides marked, A, B, C, D, E (and drawn from O in the sense in which those sides were produced). aob. Then the exterior between the sides A and B = the And the other exterior = the 2 bọc, cod, doe, eoa, respectively. ..the sum of the ext. ¿the sum of the L at O EXERCISES. 1. If one side of a regular hexagon is produced, shew that the exterior angle is equal to the interior angle of an equilateral triangle. 2. Express in degrees the magnitude of each exterior angle of (i) a regular octagon, (ii) a regular decagon. 3. How many sides has a regular polygon if each exterior angle is (i) 30°, (ii) 24°? 4. If a straight line meets two parallel straight lines, and the two interior angles on the same side are bisected, shew that the bisectors meet at right angles. 5. If the base of any triangle is produced both ways, shew that the sum of the two exterior angles minus the vertical angle is equal to two right angles. 6. In the triangle ABC the base angles at B and C are bisected by A BO and CO respectively. Shew that the angle BOC=90° + 2 7. In the triangle ABC, the sides AB, AC are produced, and the exterior angles are bisected by BO and CO. Shew that the angle 8. The angle contained by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining angles. 9. A is the vertex of an isosceles triangle ABC, and BA is produced to D, so that AD is equal to BA; if DC is drawn, shew that BCD is a right angle. 10. The straight line joining the middle point of the hypotenuse of a right-angled triangle to the right angle is equal to half the hypotenuse. EXPERIMENTAL PROOF OF THEOREM 16. [A+B+C=180°.] In the ABC, AD is perp. to BC the greatest side. AD is bisected at right angles by ZY; and YP, ZQ are perps. on BC. If now the A is folded about the three dotted lines, the LA, B, and C will coincide with the 3 ZDY, ZDQ, YDP; .. their sum is 180°. N D P 나... THEOREM 17. [Euclid I. 26.] If two triangles have two angles of one equal to two angles of the other, each to each, and any side of the first equal to the corresponding side of the other, the triangles are equal in all respects. also let the side BC= the corresponding side EF. It is required to prove that the ABC, DEF are equal in all respects. Proof. The sum of the A, B, and C Apply the BC along EF. ABC to the ▲ DEF, so that B falls on E, and .. the point A, which falls both on ED and on FD, must coin cide with D, the point in which these lines intersect ..the ABC coincides with the A DEF, and is therefore equal to it in all respects. So that AB DE, and AC DF; and the AABC= the DEF in area. Q.E.D. |