ON QUADRILATERALS. 122. DEF. A Quadrilateral is a plane figure bounded by four straight lines. 123. DEF. A Trapezium is a quadrilateral which has no two sides parallel. 124. DEF. A Trapezoid is a quadrilateral which has two sides parallel. 125. DEF. A Parallelogram is a quadrilateral which has its opposite sides parallel. TRAPEZIUM, TRAPEZOID. PARALLELOGRAM. 126. DEF. A Rectangle is a parallelogram which has its angles right angles. 127. DEF. A Square is a parallelogram which has its angles right angles, and its sides equal. 128. DEF. A Rhombus is a parallelogram which has its sides equal, but its angles oblique angles. 129. DEF. A Rhomboid is a parallelogram which has its angles oblique angles. The figure marked parallelogram is also a rhomboid. RECTANGLE. SQUARE. RHOMBUS 130. DEF. The side upon which a parallelogram stands, and the opposite side, are called its lower and upper bases; and the parallel sides of a trapezoid are called its bases. 131. DEF. The Altitude of a parallelogram or trapezoid is the perpendicular distance between its bases. 132. DEF. The Diagonal of a quadrilateral is a straight line joining any two opposite vertices. PROPOSITION XXXVIII. THEOREM. 133. The diagonal of a parallelogram divides the figure into two equal triangles. Let ABC E be a parallelogram, and AC its diagonal. ..A ABC=A A EC, $ 107 (having a side and two adj. As of the one equal respectively to a side and two adj. & of the other). Q. E. D. 134. In a parallelogram the opposite sides are equal, and the opposite angles are equal. B A E We are to prove BC= A E, and A B = EC, also, ZB= Z E, and Z BA E= ZBCE. Draw A C. § 133 A ABC=A A EC, ... B C = AE, and AB=CE LB=LE, LBAC=LACE, and LEAC= LACB, (being homologous & of equal A). Add these last two equalities, and we have ZBAC + LEAC= LACE + LACB; or, ZBA E = LBC E. Q. E. D. 135. COROLLARY. Parallel lines comprehended between parallel lines are equal. 136. If a quadrilateral have two sides equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram. Let the figure A B CE be a quadrilateral, having the side A E equal and parallel to BC. § 68 Draw A C. Нур. Iden. (being alt.-int. E). $ 106 (having two sides and the included 2 of the one equal respectively to two sides and the included 2 of the other). .. A B (being homologous sides of equal A). ZBAC=LACE, $ 69 (when two straight lines are cut by a third straight line, if the alt.-int. Is be equal the lines are parallel). $ 125 (the opposite sides being parallel). EC, Q. E. D. 137. If in a quadrilateral the opposite sides be equal, the figure is a parallelogram. Let the figure A B C E be a quadrilateral having BC= A E and AB= EC. AC=AC, Iden. ..A ABC=A AEC, (having three sides of the one equal respectively to three sides of the other). .. ZACB=LCAE, § 108 LACE, and Z BAC .. B C is || to A E, $ 69 (when two straight lines lying in the same plane are cut by a third straight line, if the alt.-int. Es be equal, the lines are parallel). § 125 .. the figure A B C E is a o, Q. E. D. |