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 ABCE be a parallelogram, and AC its diagonal. PROPOSITION XXXIX. THEOREM. 134. In a parallelogram the opposite sides are equal, and the opposite angles are equal. Let the figure A B C E be a parallelogram. (the diagonal of a divides the figure into two equal ▲). § 133 and = AB CE, = LEAC LACB, (being homologous & of equal ▲). Add these last two equalities, and we have 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. Let the figure ABCE be a quadrilateral, having the side A E equal and parallel to BC. We are to prove A B equal and || to E C. (having two sides and the included of the one equal respectively to two sides (when two straight lines are cut by a third straight line, if the alt.-int. 4 be equal the lines are parallel). .. the figure ABC E is a □, $125 Q. E. D. PROPOSITION XLI. THEOREM. 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 (having three sides of the one equal respectively to three sides of the other). (when two straight lines lying in the same plane are cut by a third straight line, if the alt.-int. ▲ be equal, the lines are parallel). .. the figure ABCE is a □, $125 Q. E. D. |