PROPOSITION XXXIV. THEOREM 159. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. B Given the quadrilateral ABCD in which AB=CD_and BC=AD. (if two lines are cut by a transversal, making the alt. int. 4 equal, the Also lines are ). LBCA = L CAD. :. BC|| AD. .. ABCD is a, (Why ?) Art. 125. Art. 147. (a is a quadrilateral whose opposite sides are ||). Q. E. D. PROPOSITION XXXV. THEOREM 160. If two sides of a quadrilateral are equal and parallel, the other two sides are equal and parallel and the figure is a parallelogram. B Given the quadrilateral ABCD in which BC = and | AD. (if two lines are cut by a transversal, making the alt. int. & equal, the lines are ). :. ABCD is a Art. 147. Q. E. D. Ex. 1. Show that in a each pair of adjacent angles is supplementary. Ex. 2. One angle of a parallelogram is 43°; find the other angles. Ex. 3. If, in the triangle ABC, LA=60°, ≤B=70°, which is the longest side in the triangle? Which the shortest ? PROPOSITION XXXVI. THEOREM 161. The diagonals of a parallelogram bisect each other. F Given the diagonals AC and BD of the ABCD, intersecting at F. To prove AF-FC, and BF-FD. Proof. Let the pupil supply the proof. [SUG. In the ▲ BFC and AFD what sides are equal, and why? What are equal, and why? etc.] Q. E. D. Ex. 1. How many pairs of equal triangles are there in the above figure? Ex. 2. If one angle of a parallelogram is three times another angle, find all the angles of the parallelogram. Ex. 3. If two angles of a triangle are po and q°, find the third angle. Ex. 4. If two angles of a triangle are x and 90° + x°, find the third angle. Ex. 5. If one angle of a parallelogram is a°, find the other angles. Ex. 6. Construct exactly an angle of 60°. Ex. 7. How large may the double of an acute angle be? how small? Ex. 8. How large may the double of an obtuse angle be? how small? PROPOSITION XXXVII. THEOREM 162. 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 ABCD and A'B'C'D' in which AB=A'B', AD=A'D', and ▲A = ZA'. Given the To prove ABCD = ☐ A'B'C'D'. Proof. Apply the A'B'C'D' to the ABCD so that A'D' shall coincide with its equal AD. Then A'B' will take the direction of AB (for LA'= LA); and point B' will fall on B (for A'B'=AB). Then B'C' and BC will both be || AD and will both pass through the point B. :. B'C' will take the direction of BC, Geom. Ax. 3. (through a given point one straight line, and only one, can be drawn || another given straight line). In like manner, D'C' must take the direction of DC. :. C' must fall on C, (two straight lines can intersect in but one point). Art. 64. Art. 47. Q. E. D. 163. COR. Two rectangles which have equal bases and equal altitudes are equal. Ex. Construct exactly an angle of 30°, POLYGONS 164. A polygon is a portion of a plane bounded by straight lines, as ABCDE. The sides of a polygon are its bounding lines; the perimeter of a polygon is the sum of its sides; the angles of a polygon are the angles formed by its sides; the vertices of a polygon are the vertices of its angles. A diagonal of a polygon is a straight line joining two vertices which are not adjacent, as BD in Fig. 1. 165. An equilateral polygon is a polygon all of whose sides are equal. 166. An equiangular polygon is a polygon all of whose angles are equal. What four-sided polygon is equilateral but not equiangular? Also, what four-sided polygon is both equilateral and equiangular? 167. A convex polygon is a polygon in which no side, if produced, will enter the polygon, as ABCDE (Fig. 1). Each angle of a convex polygon is less than two right angles and is called a salient angle. 168. A concave polygon is a polygon in which two or more sides, if produced, will enter the polygon, as FGHIJK (Fig. 2), |