Ex. 182. In the annexed diagram, if AB = AC, prove that BD > DC. Ex. 183. In the diagram of Ex. 182, prove that BE> EC. Ex. 184. In the diagram of Ex. 182, prove that AF> AB. Ex. 185. In the diagram of Ex. 182, prove that AB> AH. *Ex. 186. If the opposite sides of a quadrilateral ABCD are equal, but AB > AD, prove that LAOB>ZAOD. *Ex. 187. The sum of the lines drawn from any point in a triangle to its vertices is less than the perimeter, but greater than the semiperimeter of the triangle. Ex. 188. The sum of the diagonals of any quadrilateral is less than the perimeter, but greater than the semiperimeter of the quadrilateral. QUADRILATERALS 131. DEF. A trapezoid is a quadrilateral that has one pair of sides parallel. A parallelogram has its opposite sides parallel. A rhombus is an equilateral parallelogram, whose angles are oblique. A rectangle is a parallelogram, whose angles are right angles. A square is an equilateral rectangle. E 132. DEF. An isosceles trapezoid is one whose non-parallel sides are equal. The parallel sides of a trapezoid are called its bases, and are distinguished as upper and lower. 133. 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 XXXII. THEOREM 134. The opposite sides and angles of a parallelogram are equal. A B Нур. D ABCD is a parallelogram. To prove AD = BC; AB = CD; ZA=LC; ≤B=LD. HINT. - What is the usual means of proving the equality of lines and angles ? 135. COR. 1. A diagonal divides a parallelogram into two equal triangles. 136. COR. 2. If one angle of a parallelogram is a right angle, the figure is a rectangle. 137. COR. 3. Parallels included between parallels are equal. Ex. 189. The perpendiculars to a diagonal of a parallelogram from the opposite vertices are equal. Ex. 190. The diagonals of a parallelogram bisect each other. * Ex. 191. A line bisecting one side of a triangle and parallel to another side, bisects the third side also. 138. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. D In quadrilateral ABDC AB= CD; AD = BC. AD || BC; AB || CD. HINT. Prove the equality of a pair of alternate interior angles by means of equal triangles. Ex. 192. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. PROPOSITION XXXIV. THEOREM 139. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. Нур. To prove D In quadrilateral ABCD AB || CD; and AB= CD. ABCD is a parallelogram. [The proof is similar to that in Prop. XXXIII.] Ex. 193. The bisectors of two opposite angles of a parallelogram are parallel. Ex. 194. If two opposite sides of a parallelogram are produced by the same length in opposite directions and their ends joined to the nearest vertices, another parallelogram is formed. 140. REMARK. Lines may be shown to be parallel by proving them to be opposite sides of a parallelogram. Ex. 195. If two opposite sides of a parallelogram are divided into three equal parts, and the respective points of division are joined, the lines are parallel. Ex. 196. If two opposite sides of a parallelogram are produced by the same length in the same direction, a line joining the ends is parallel to the other sides of the parallelogram. Ex. 197. If from two opposite vertices of a parallelogram lines be drawn bisecting two opposite sides, respectively, the lines are parallel. PROPOSITION XXXV. THEOREM 141. The diagonals of a parallelogram bisect each other. B Hyp. To prove ABDC is a parallelogram. AO = OD; BO = OC. [The proof is left to the student.] Ex. 198. The diagonals of a rhombus are perpendicular to each other. Ex. 199. If the diagonals of a parallelogram are perpendicular to each other, the figure is a rhombus, or a square. Ex. 200. If each half of the diagonal of a parallelogram is bisected, and the midpoints are joined in order, another parallelogram is formed. Ex. 201. If a diagonal bisects an angle of a parallelogram, the figure is a rhombus. Ex. 202. The diagonals of a rectangle are equal. Ex. 203. State and prove the converse of the preceding exercise. Ex. 204. If the ends of two diameters of a circle be joined in succession a rectangle is formed. Ex. 205. The base angles of an isosceles trapezoid are equal. Ex. 206. State and prove the converse of the preceding exercise. PROPOSITION XXXVI. THEOREM 142. Two parallelograms are equal if two adjacent sides and the included angle of one are equal, respectively, to two adjacent sides and the included angle of the other. B A' B' D' Hyp. In ABCD and A'B'C'D', AB = A'B', AD = A'D', LA = LA'. To prove ABCD=□ A'B'C'D'. Proof. Apply□ABCD to A'B'C'D', so that AB coincides with A'B'. Then AD takes the direction A'D', (ZA = ZA'). And D coincides with D', (for AD = A'D'). BC takes the direction of B'C", and C must lie in B'C' or in B'C' produced. (Ax. 11.) For a similar reason, C must lie in D'C' or in D'C' produced. Hence and D coincides with D', ABCD coincides with A'B'C'D'. .. ABCD = Q.E.D. |