PROPOSITION XXIII. Straight lines which join the extremities of equal and parallel straight lines towards the same parts are themselves equal and parallel. Join BC. and ::: in the As ABC, DCB, = DC, CB, and .DCB; Hence AC is = DB. Also - ACB is = 1 DBC, and ... AC is || to BD. (1. 20) PROPOSITION XXIV. If one side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; also the three interior angles of every triangle are together equal to two right angles. 25 Let ABC be a a, and let one of its sides BC be produced to K. Then shall LACK be equal to the anglès ABC, BAC together; also the three interior angles of the A ABC shall be together equal to two right angles. Through C draw CH | to AB. Then :: CH is I to AB, and AC meets them, therefore LACH is = the alternate .BAC. (1. 21) Again, :;: CH is || to AB, and BK falls upon them ; .. the exterior .HCK is = the interior and opposite LABC; (1. 21) ... the whole LACK is = the two angles ABC, BAC together. Now to each of these equals add the LACB; .. the two angles ACK, ACB are together = the three es of the A ABC; but the two angles ACK, ACB are together = two right 2 s; . (1. 9) is the three bs of ABC are together = two right < S. PROPOSITION XXV. If two angles of one triangle are equal to two angles of another, then shall the remaining angle of the one be equal to the remaining angle of the other. For the three angles of each a are together = two right 2 s; (I. 24) .. the three angles of the one are together = the three Zs of the other; but two zs of the one are = two zs of the other; .. the remaining % of the one is = the remaining of the other. Cor. If two angles of one triangle are equal to two angles of another, and any side of one is = the corresponding side of the other; then shall the as be equal in all respects. For, by the proposition, the remaining zs are = one another. Hence there are two angles and the side adjacent in the one triangle equal to two angles and the side adjacent in the other; .. the as are equal in all respects. (1. 3) QUADRILATERAL AND MULTILATERAL FIGURES. DEFINITIONS. Rectilineal figures are those contained by straight lines. Quadrilateral figures by four straight lines. Multilateral figures, or polygons, by more than four straight lines. A parallelogram is a four-sided figure, of which the opposite sides are parallel. The straight line joining the opposite angles of a quadrilateral figure is called a diagonal. PROPOSITION XXVI. The opposite sides and angles of a parallelogram are equal, and the diagonal bisects it. Let ABCD be a D, and Then shall AB be = CD, PROPOSITION XXVII. If one angle of a parallelogram is a right angle, all its angles are right angles. Let PQRS be a , having the 2 PSR a right 2. DEFINITION. A rectangle is a right-angled parallelogram, and is said to be contained by any two of its adjacent sides. |