consistent, and that Euclid's system, though undoubtedly the most simple, is only one of many systems that could be devised. Till 1915 no practical application was made of these alternative systems; but Einstein's theory of the universe questions whether the geometry of the space in which we live is Euclidean Geometry. THEOREM 5 [CONVERSE OF TH. 4]. If a straight line cuts two parallel straight lines, (i) alternate angles are equal, (ii) corresponding angles are equal, (iii) the interior angles on the same side of the cutting line are together equal to two right angles. .. the two straight lines PG, CG which pass through the point NOTE. Any one of (i), (ii), (iii) can be proved by Playfair's Axiom and the other two deduced from it. ¶Ex. 4. Prove Th. 5 (ii) by Playfair's Axiom and deduce Th. 5 (i) and (iii). ¶Ex. 5. Prove Th. 5 (iii) by Playfair's Axiom and deduce Th. 5 (i) and (ii). THEOREM 6. Straight lines which are parallel to the same straight line are parallel to one another. Construction Draw a st. line cutting AB, CD, XY and forming with them corresponding s p, q, z respectively. Proof [We shall prove that ▲p=Lq.] THEOREM 7. If straight lines are drawn from a point parallel to the arms of an angle, the angle between those straight lines is equal or supplementary to the given angle. From O, OX is drawn || to AB and in the same sense and OY is drawn || to AC and in the same sense To prove that Proof * AB and BA are said to be drawn in opposite senses. AB and OX in fig. 8 are and in the same sense; AB and OZ are || and in opposite senses. ANGLE-SUM for TrianGLE AND POLYGON. THEOREM 8. The sum of the angles of a triangle is equal to two right angles. Data To prove that ABC is a triangle. LA+LB+ 4 BCA = 2 rt. 4 s. Produce BC to D. Through C draw CE || to BA. Proof [We shall prove that ▲s of ▲ = Ls at C.] D COR. 1. If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles; and therefore an exterior angle is greater than either of the two interior opposite angles. (Proof as above.) COR. 2. If two triangles have two angles of the one equal to two angles of the other, each to each, then the third angles are also equal. COR. 3. The sum of the angles of a quadrilateral is equal to four right angles. (Draw a diagonal.) COR. 4. In a right-angled triangle (i) the right angle is the greatest angle; (ii) the sum of the remaining angles is equal to a right angle. THEOREM 9. If the sides of a convex polygon are produced in order, the sum of the angles so formed is equal to four right angles. Data ABCDE is a convex polygon; its sides are produced in order and form the exterior angles w, v, x, y, z. To prove that 4 rt. 4 s. LW + LV + L x + LY + L2 = Construction Through any point O draw OP, OQ, OR, OS, OT || to and in the same sense as EA, AB, BC, CD, DE respectively. Proof Since OP, OQ are respectively || to and in the same sense as EA, AB :: LW + LV + L x + L Y + L Z = sum of 4 s at O COR. The sum of the interior angles of any convex polygon together with four right angles is equal to twice as many right angles as the polygon has sides. [For the interior and exterior angles at n vertices = n × 2 rt. ≤ s, .. the interior angles (2n-4) rt. s.] = Some writers and schedules give the corollary the greater prominence; but for actual use the theorem is much the more useful. |