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. В Let ABC E be a parallelogram, and A C its diagonal. We are to prove A ABC=A A EC. Iden. (being alt.-int. 4). § 68 .. A ABC=A AEC, § 107 (having a side and two adj. As of the one equal respectively to a side and two adj. s of the other). Q. E. D. § 68 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. and We are to prove BC= A E, and A B = EC, Draw A C. § 133 (the diagonal of a divides the figure into two equal A). ..BC=A E AB=CE, ZB=LE, Z BAC = LACE, Z EAC = L ACB, (being homologous ts of equal A). Z BAC + < EAC = LACE + ZACB; Q. E. D. 135. COROLLARY. Parallel lines comprehended between parallel lines are equal. 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. Hyp. Let the figure A BCE be a quadrilateral, having the side A E equal and parallel to BC. Draw A C. BC = A E, Iden. $ 68 (being alt.-int. E). .. A ABC=A ACE, § 106 Chaving two sides and the included L of the one equal respectively to two sides and the included L of the other). . AB= EC, (being homologous sides of equal A ). < BAC = ZAC E, $ 69 (when two straight lines are cut by a third straight line, if the alt.-int. A 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. В Hyp. Let the figure A B C E be a quadrilateral having BC= AE and AB= EC. Draw A C. B = 4 E, Hyp. Iden. ..A ABC=A A EC, $ 108 (having three sides of the one equal respectively to three sides of the other). .. ZACB=LCA E, ZBAC=LACE, ..BC is to A E, A B is I to EC, (when two straight lines lying in the same plane are cut by a third straight line, if the alt.-int. E be equal, the lines are parallel). $ 125 (having its opposite sides parallel). Q. E. D. and § 69 |