then the included angle of the first is greater than the included angle of the second. [Converse of Prop. XXXI.] Given A ABC and A'B'C', with bb'; c=c'; a>a. 134. NOTE. In the demonstrations which follow, the reasons will, as a rule, not be given in detail, but merely by reference to the paragraph number. The student, however, should in many cases state the proofs completely, with reasons in full, although occasionally the mere "statements" will be sufficient to test the pupil's understanding. Ex. 328. If in ▲ ABC, having AC> BC, the median CE is drawn, then ZAEC is an obtuse angle. Ex. 329. Given ▲ ABC, having ZA = L B. If the point D in AB be taken so that AD> DB, Á then ACD > LDCB. B 135. METHOD VIII. Inequality of lines is generally proved by means of Props. XXVIII, XXIX, XXXI. If it is impossible to discover any relation of the angles, Prop. XXVIII is used; if the two lines are parts of the same triangle, and the opposite angles can be proved unequal, Prop. XXIX is used; but if the sides or angles are parts of different triangles, Prop. XXXI is used. In some cases several methods will lead to the desired result. The inequality of angles is proved in a similar manner by Props. XII, XXX, and Prop. XXXII. Ex. 333. In the annexed diagram, if AB = AC, prove that (a) BD> DC. *E (b) BE> EC. (c) AF> AB. (d) AB>AH. Ex. 334. The sum of the diagonals of any quadrilateral is less than the perimeter, but greater than the semiper. imeter of the quadrilateral. * Ex. 337. If in ▲ ABC, having AB< BC, BD, the bisector of angle B, is drawn, meeting AC in D, then AD < DC. * Ex. 338. If in ▲ ABC, having AC> BC, the median CD is drawn, then any point E in BD is nearer to B than to A. QUADRILATERALS 136. DEF. A trapezium is a quadrilateral having no two sides parallel. A trapezoid is a quadrilateral having two, and only two, sides parallel. A parallelogram is a quadrilateral having its opposite sides. parallel. TRAPEZIUM TRAPEZOID PARALLELOGRAM 137. DEF. A rectangle is a parallelogram whose angles are right angles. A rhomboid is a parallelogram whose angles are oblique. 138. DEF. A square is an equilateral rectangle. A rhombus is an equilateral rhomboid. 139. DEF. An isosceles trapezoid is a trapezoid whose nonparallel sides are equal. The parallel sides of a trapezoid are called its bases, and are distinguished as upper and lower. The median of a trapezoid is the line joining the mid-points of the nonparallel sides. 140. DEF. A diagonal of a quadrilateral is a straight line joining opposite vertices. The altitude of a parallelogram or trapezoid is the perpendicular distance between the two bases. PROPOSITION XXXIII. THEOREM 141. The opposite sides and angles of a parallelogram are equal. To prove AD = BC; AB = CD; ▲ A = ≤C; B = D. HINT. What is the usual means of proving the equality of lines and angles? 142. COR. 1. A diagonal divides a parallelogram into two congruent triangles. 143. COR. 2. If one angle of a parallelogram is a right angle, the figure is a rectangle. Nek Cox & Parallels included between parallels are equal. Na Cox & If two adjacent sides of a parallelogram are qua, e Sgure is equilateral, and hence either a rhombus or a The perpendiculars to a diagonal of a parallelogram from the Pas vertires are equal. $40. Let ABCD be a parallelogram and BE and DF lines drawn perpendicular to AD and BC. Prove BE = DF. X941. If in parallelogram ABCD the diagonals meet in E, then 03-300 IX $42. Let ABCD be a parallelogram and BD a diagonal, prove sha the angle-bisectors AE and CF of triangles ABD and CBD respec ey are equal. x. $43. If the points G and E trisect the diagonal BD of parallelogem ABCD, prove that AG = CE. Ex. $44. The diagonals of a rectangle are equal. Ex. 345. The diagonals of a rhomboid are unequal. Ex. 346. If the diagonals of a parallelogram are equal, the figure is a rectangle. PROPOSITION XXXIV. THEOREM 146. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. HINT. Prove the equality of a pair of alternate interior angles by eans of congruent triangles. |