DRILATERALS. 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 parallelogranı 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. в Let A B C E be a parallelogram, and A C its diagonal. AC=AC, Iden. ZACB= 2 CAE, $ 68 (being alt.-int. £). Z CAB=LACE, § 68 ... A ABC=A A EC, § 107 (having a side and two adj. E of the one equal respectively to a side and two adj. É of the other). Q. E. D. PROPOSITION XXXIX. THEOREM. 134. In a parallelogram the opposite sides are equal, and the opposite angles are equal. B . and Let the figure A B C E be a parallelogram. Draw A C. $ 133 (the diagonal of a divides the figure into two equal A). .:. BC = A E, AB=CE, ZB=LE, ZBAC = LACE, Z EAC = Z A C B, (being homologous ts of equal A). ZBAC + 2 E AC=LACE + LACB; Q. E. D. 135. COROLLARY. Parallel lines comprehended between parallel lines are equal. and PROPOSITION XL. THEOREM. 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. B Let the figure A BCE be a quadrilateral, having the side A E equal and parallel to B C. Draw A C. Hyp. Iden. ZBCA = LCA E, § 68 (being alt.-int. 4). .. A ABC = A ACE, § 106 (having two sides and the included L of the one equal respectively to two sides and the included 2 of the other). .. A B = EC, (being homologous sides of equal A). < BAC = LACE, $ 69 (when two straight lines are cut by a third straight line, if the alt. -int. És be equal the lines are parallel). § 125 (the opposite sides being parallel). Q. E. D. PROPOSITION XLI. THEOREM. 137. If in a quadrilateral the opposite sides be equal, the figure is a parallelogram. Hyp. Let the figure A B C E be a quadrilateral having BC = A E and A B = EC. Draw A C. Hyp. Iden. .:. A ABC=A AEC, § 108 (having three sides of the one equal respectively to three sides of the other). Z ACB= 2 CAE, ZBAC=LACE, .. B C is I to A E, § 69 (when two straight lines lying in the same plane are cut by a third straight line, if the alt.-int. Is be equal, the lines are parallel). § 125 (having its opposite sides parallel). Q. E. D. |