6. Trisect a right angle. 7. The angle contained by the bisectors of the angles at the base of an isosceles triangle is equal to an exterior angle formed by producing the base. 8. The angle contained by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining angles. The following theorems were added as corollaries to Proposition 32 by Robert Simson, who edited Euclid's text in 1756. 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. Let ABCDE be any rectilineal figure. Take F, any point within it, and join F to each of the angular points of the figure. Then the figure is divided into as many triangles as it has sides. And the three angles of each triangle are together equal to two right angles. I. 32. Hence all the angles of all the triangles are together equal to twice as many right angles as the figure has sides. But all the angles of all the triangles make up all the interior angles of the figure, together with the angles at F, which are equal to four right angles. 1. 15, Cor. Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides. Q.E.D. 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. For at each angular point of the figure, the interior angle and the exterior angle are together equal to two right angles. I. 13. Therefore all the interior angles, with all the exterior angles, are together equal to twice as many right angles as the figure has sides. But all the interior angles, with four right angles, are together equal to twice as many right angles as the figure has sides. 1. 32, Cor. 1. Therefore all the interior angles, with all the exterior angles, are together equal to all the interior angles, with four right angles. Therefore the exterior angles are together equal to four right angles. Q.E.D. EXERCISES ON SIMSON'S COROLLARIES. [A polygon is said to be regular when it has all its sides and all its angles equal.] 1. Express in terms of a right angle the magnitude of each angle of (i) a regular hexagon, (ii) a regular octagon. 2. If one side of a regular hexagon is produced, shew that the exterior angle is equal to the angle of an equilateral triangle. 3. Prove Simson's first Corollary by joining one vertex of the rectilineal figure to each of the other vertices. 4. Find the magnitude of each angle of a regular polygon of n sides. 5. If the alternate sides of any polygon be produced to meet, the sum of the included angles, together with eight right angles, will be equal to twice as many right angles as the figure has sides. PROPOSITION 33. THEOREM. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel. D Let AB and CD be equal and parallel straight lines ; and let them be joined towards the same parts by the straight lines AC and BD. Then shall AC and BD be equal and parallel. Proof. Then because AB and CD are parallel, and BC meets them, therefore the angle ABC is equal to the alternate angle DCB. I. 29. Hyp. Because and BC is common to both; also the angle ABC is equal to the angle DCB; Proved. therefore the triangle ABC is equal to the triangle DCB in all respects; I. 4. But these are alternate angles. I. 27. and it has been shewn that they are also equal. Q.E.D. DEFINITION. A Parallelogram is a four-sided figure whose opposite sides are parallel. PROPOSITION 34. THEOREM. The opposite sides and angles of a parallelogram are equal to one another, and each diagonal bisects the parallelogram. А B Let ACDB be a parallelogram, of which BC is a diagonal. Then shall the opposite sides and angles of the figure be equal to one another ; and the diagonal BC shall bisect it. Proof. Because AB and CD are parallel, and BC meets them, therefore the angle ABC is equal to the alternate angle DCB; I. 29. Again, because AC and BD are parallel, and BC meets them, therefore the angle ACB is equal to the alternate angle DBC. I. 29. Hence in the triangles ABC, DCB, (the angle ABC is equal to the angle DCB, Because and the angle ACB is equal to the angle DBC; also the side BC is common to both; therefore the triangle ABC is equal to the triangle DCB in all respects; I. 26. so that AB is equal to DC, and AC to DB; and the angle BAC is equal to the angle CDB. Also, because the angle ABC is equal to the angle DCB, and the angle CBD equal to the angle BCA, therefore the whole angle ABD is equal to the whole angle DCA. And the triangles ABC, DCB having been proved equal in all respects are equal in area. Therefore the diagonal BC bisects the parallelogram ACDB. Q.E.D. EXERCISES ON PARALLELOGRAMS. 1. If one angle of a parallelogram is a right angle, all its angles are right angles. 2. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. 3. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. 4. If a quadrilateral has all its sides equal and one angle a right angle, all its angles are right angles. 5. The diagonals of a parallelogram bisect each other. 6. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 7. If two opposite angles a parallelogram are bisected by the diagonal which joins them, the figure is equilateral. 8. If the diagonals of a parallelogram are equal, all its angles are right angles. 9. In a parallelogram which is not rectangular the diagonals are unequal. 10. Any straight line drawn through the middle point of a diagonal of a parallelogram and terminated by a pair of opposite sides, is bisected at that point. 11. If two parallelograms have two adjacent sides of one equal to two adjacent sides of the other, each to each, and one angle of one equal to one angle of the other, the parallelograms are equal in all respects. 12. Two rectangles are equal if two adjacent sides of one are equal to two adjacent sides of the other, each to each. 13. In a parallelogram the perpendiculars drawn from one pair of opposite angles to the diagonal which joins the other pair are equal. 14. If ABCD is a parallelogram, and X, Y respectively the middle points of the sides AD, BC; shew that the figure AYCX is a parallelogram. |