PROP. XXXIII. THEOREM 107. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. B A Draw line BC = and || to AD, and lines AB and DC. We then have: (Prove ABC and ACD equal by § 46; then, AB = CD. Compare § 106.) PROP. XXXIV. THEOREM 108. The diagonals of a parallelogram bisect each other. Draw ABCD as in § 107; and lines AC and BD. We then have : Given diagonals AC and BD of ABCD intersecting at E. To Prove AE EC and BE = (Prove A AED = A BEC, by § 49.) = ED. Note. The point E is called the centre of the parallelogram. PROP. XXXV. THEOREM 109. (Converse of Prop. XXXIII.) If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. Given AC and BD, the diagonals of quadrilateral ABCD, bisecting each other at E. (Fig. of Prop. XXXIV.) To Prove ABCD a □. (Prove ▲ AED ▲ BEC, by § 46; then AD = BC ; manner, AB = = CD; then use § 106.) in like PROP. XXXVI. THEOREM 110. Two parallelograms are equal when two adjacent sides and the included angle of one are equal respectively to two adjacent sides and the included angle of the other. Draw ABCD and EFGH in accordance with the statement of the proposition. Let a, b, c, d, represent sides AB, BC, CD, and DA, respectively; and e, f, g, h, sides EF, FG, GH, and HE, respectively. We then have : way that ▲ ▲ shall coincide with its equal E; side a falling on side e, and side d on side h. 2. Since ae, point B will fall on point F. 3. Since d=h, point D will fall on point H. 4. Since bd, and fh, side b will fall on side f, and point C will fall somewhere on f. [But one str. line can be drawn through a given point || to a given str. line.] ($ 69) 5. Since cla, and g | e, side c will fall on side g, and point C will fall somewhere on g. 6. Since C falls on both ƒ and g, it must fall at their intersection, G. 7. Then, the coincide throughout, and are equal. The following is an immediate consequence of § 110: 111. Two rectangles are equal if the base and altitude of one are equal respectively to the base and altitude of the other. PROP. XXXVII. THEOREM 112. The diagonals of a rectangle are equal. Draw rectangle ABCD; draw lines AC and BD. We then have: (Prove rt. ▲ ABD = rt. ▲ ACD, by § 46.) The following is an immediate consequence of § 112: 113. The diagonals of a square are equal. 114. The diagonals of a rhombus bisect each other at right angles. B C E A Given AC and BD the diagonals of rhombus ABCD. To Prove that AC and BD bisect each other at rt. . (Compare § 56.) Ex. 37. The lines which join the middle points of the opposite sides of a parallelogram bisect each other. Ex. 38. Are the diagonals of a parallelogram ever equal? Ex. 39. When do the diagonals of a parallelogram bisect the opposite angles? Ex. 40. If perpendiculars be drawn from the extremities of the upper base of an isosceles trapezoid to the lower base, they are equal and cut off equal segments from the extremities of the lower base. Ex. 41. The lines which join the middle points of the bases of an isosceles trapezoid to the middle points of the non-parallel sides form, with the sides of the trapezoid, two pairs of equal triangles. Ex. 42. If lines be drawn from the middle point of the base of an isosceles triangle to the middle points of the equal sides, the triangle is divided into two equal triangles and a parallelogram. POLYGONS DEFINITIONS 115. We define a polygon as a portion of a plane bounded by three or more straight lines; as ABCDE. We call the bounding lines the sides of the polygon, and their sum the perimeter. The angles of the polygon are the A angles EAB, ABC, etc., formed by the adjacent sides; their vertices are called the vertices of the polygon. B E A diagonal of a polygon is a straight line joining any two vertices which are not consecutive; as AC. 116. Polygons are named with reference to the number of their sides, as follows: 117. An equilateral polygon is a polygon all of whose sides are equal. An equiangular polygon is a polygon all of whose angles are equal 118. A polygon is called convex when no side, if extended, will enter its surface; as ABCDE; in such a case, each angle of the polygon is less than two right angles. Every polygon considered hereafter will be understood to be convex, unless the contrary is stated. 119. A polygon is called concave when at least two of its sides, if extended, will enter its surface; as FGHIK. In such a case, at least one angle of the polygon is greater than two right angles. Thus, in polygon FGHIK, the interior angle GHI is greater than two right angles; such an angle is called re-entrant. 120. We call two polygons mutually equilateral when the sides of one are equal re spectively to the sides of the other, when taken in the same order. B B Thus, polygons ABCD and Α' D' A'B'C'D' are mutually equilateral if AB A'B', BC= B'C', CD = C'D', and DA = D'A'. = 121. We call two polygons mutually equiangular when the angles of one are equal respec G ZA=ZE, ZB = LF, LC = Z G, and ≤ D = 2 H. 122. In polygons which are mutually equilateral or mutually equiangular, sides or angles which are similarly placed are called homologous. |