Also AFG is equal to EFB (Th. 3), and therefore equal to FGD, which is its alternate angle. And BFG is supplementary to EFB (Th. 1), and therefore also to FGD. THEOREM 5. If two straight lines meet in a point they will make unequal corresponding angles with every straight line which meets them. Let AO and BO meet at O, and be intersected by any straight line CD, in the points A and B. Then will the corresponding angles B CAO, CBO be unequal. D A O' Proof. Then since AO and BO meet they have different directions (Ax. 7), and therefore if the angles B were transferred to A, the direction of their arms being unaltered, the arm BO would not coincide with AO, while BA would coincide with AC. Hence the angle CAO is not equal to the corresponding angle CBO. COR. I. Hence if two straight lines which are not parallel are intersected by a third, the alternate angles will be not equal, and the interior angles on the same side of the intersecting line will be not supplementary. COR. 2. Hence also if the corresponding angles are equal, or the alternate angles equal, or the interior angles supplementary, the lines will be parallel. For they cannot be not parallel, for then the corresponding and alternate angles would be unequal by Cor. 1. COR. 3. From a given point outside a given line only one perpendicular can be let fall on that line. Remark. Theorems may often be arranged in groups of four; an original theorem, its opposite, its converse, and the converse of its opposite. Thus, "If A is B, then C will be D," may be taken as the type of a theorem. "If A is B" is the hypothesis or supposition; " C will be D" is the conclusion. The opposite of a theorem is formed by negativing the hypothesis and conclusion. Thus, "If A is not B, then C will not be D," is the opposite of the original theorem. The converse of a theorem is formed by transposing the hypothesis and conclusion. Thus, "If C is D, then A will be B" is the converse. And obviously, "If C is not D, then A will not be B" is the converse of the opposite, or the opposite of the converse. It must be observed that if two of these theorems are true, then the rest follow logically. Thus, "If two lines are parallel, the corresponding angles will be equal" is the original theorem; its opposite is, "If two lines are not parallel, the corresponding angles will be unequal." Hence the converse theorem follows that "if the corresponding angles are equal the lines will be parallel," and the converse of the opposite, that "if the corresponding angles are unequal the lines will not be parallel." Def. 21. A figure enclosed by any number of straight lines is called a polygon. It is called convex when no one of its angles is reflex. It is called regular when it is equilateral and equiangular, that is when all its sides and angles are equal. A regular four-sided figure is called a square. The line joining any two angles not adjacent is called a diagonal. When the number of its sides is 3, 4, 5, 6..., it is called a triangle, a quadrilateral, a pentagon, a hexagon, and so on. THEOREM 6. The exterior angles of any convex polygon, made by producing the sides in succession, will be together equal to four right angles. Let ABCDE be a polygon having all its sides produced in succession; then will the sum of its exterior angles be equal to four right angles. Proof. For conceive lines drawn through any point O parallel to the sides of the polygon, in the directions in which the sides are produced. Then (by Theorem 4) the angles taken consecutively round O are equal to the exterior angles of the polygon. But the angles at O are together equal to four right angles (Th. 2). Therefore the exterior angles of the polygon are together equal to four right angles. COR. Hence it may be shewn that all the interior angles of any polygon are less than twice as many right angles as the figure has sides by four right angles. Proof. For each interior angle with its adjacent exterior angle two right angles (Th. 1); = Therefore all the interior angles + all the exterior angles twice as many right angles as the figure has sides; But all the exterior angles four right angles (Th. 6); = Therefore all the interior angles + four right angles twice as many right angles as the figure has sides; and therefore all the interior angles are less than twice as many right angles as the figure has sides by four right angles. This Corollary leads to many interesting results. Let it be required, for example, to find the magnitude of the angle of a regular hexagon. Since the six interior angles are less than twelve right angles by four right angles, they are together equal to eight right angles; and therefore each of them is eight-sixths of a right angle, or is of a right angle, or contains 120o. I. EXERCISES. Prove that the angles of any triangle are together equal to two right angles. 2. Shew that the angles of an equiangular triangle are equal to two-thirds of a right angle. Express them in degrees. 3. Find the magnitude of the angle of a regular octagon. 4. How many equiangular triangles can be placed so as to have one common angular point, and fill up the space round it? 5. Shew that three regular hexagons can be placed so as to have a common point, and fill up the space round that point. 6. Shew that two regular octagons and one square have the same property. Draw a pattern consisting of octagons and squares. 7. Shew that the angle of a regular pentagon is to the angle of a regular decagon as 3 to 4. W. G. с 8. If a line is perpendicular to another it will be perpendicular to every line parallel to it. 9. If a polygon is equilateral, does it follow that it is equiangular, and conversely? IO. Shew that the exterior angle of a regular polygon of I n sides is of 360°: and that its interior angle is II. n angle is (2–4)90o. How many diagonals can be drawn in a pentagon? How many in a decagon ? 12. Shew that a square, a hexagon and a dodecagon will fill up the space round a point; and make a pattern of these polygons. 13. Examine whether a square, a pentagon and an icosagon have the same property; and also whether a pattern can be constructed of pentagons and decagons. 14. The exterior angle of a regular polygon is one-third of a right angle: find the number of sides in the polygon. 15. Two lines intersecting in A are respectively perpendicular to two lines intersecting in B: prove that any angle at A is equal or supplementary to any angle at B. 16. The arms of an angle A are equally inclined, in the same direction of rotation, to the arms of an angle B. Prove that the angle A is equal to the angle B. |