But the angles at A and at B together are half of the four. Therefore the angles at A and at B together make up two right angles. But the angles at A and at B are interior angles with respect to the two lines DA, CB and are on the same side of the line AB. Therefore DA and CB are parallel. Similarly it may be proved that AB, DC are parallel. So that ABCD is a parallelogram. Q.E.D. Example ii. The straight line joining the middle points of two sides of a triangle is parallel to the base. Let E, F be the middle points of the sides AC, AB of the triangle ABC; it is required to prove that EF is parallel to the base BC. we proceed to prove that EG and EF are coincident. GEDB is a parallelogram; therefore DE=BG. [Prop. 34.] Again, because DE and GA A therefore the angles ECD, AEG are equal. [Prop. 29.] Now, consider the triangles AGE, EDC; because the side AE and the angles GAE, AEG are equal respectively to the side EC and the angles DEC, ECD, therefore, by Prop. 26 (I), the triangles AEG, ECD are equal in all respects. Therefore BG=GA; that is, G coincides with F, the middle point of AB; so that EG and EF are coincident. Wherefore EF is parallel to BC. Q.E.D. EXAMPLES XXXIX. 1. The diagonals of a parallelogram bisect each other. 2. A quadrilateral whose diagonals bisect each other is a parallelogram. 3. A quadrilateral whose opposite sides are equal is a parallelogram. 4. A diagonal of a rhombus bisects two of its angles. 5. When one angle of a parallelogram is bisected by a diagonal, it is a rhombus. 6. If one angle of a parallelogram is a right angle all its angles are right angles. 7. The diagonals of a rectangle are equal. 8. If the diagonals of a parallelogram are equal it is a rectangle. 9. All the sides of a square are equal. 10. All the angles of a square are right angles. 11. The diagonals of a square are equal. 12. The diagonals of a square bisect each other at right angles. 13. If the diagonals of a quadrilateral bisect each other at right angles the quadrilateral is a rhombus. 14. If the diagonals of a rhombus are equal it is a square. 15. If the diagonals of a quadrilateral are equal and bisect each other at right angles the figure is a square. 16. The diagonals of a parallelogram are unequal except when it is a rectangle. 17. Every straight line drawn through the point of intersection of the diagonals of a parallelogram and terminated by a pair of opposite sides is bisected at that point. 18. If one angle of a parallelogram is equal to one angle of another all the angles of the first parallelogram are equal respectively to the angles of the second. 19. If two sides of a parallelogram be produced from their point of intersection the four angles at this point are equal to the angles of the parallelogram. 20. If two adjacent sides and an angle of one parallelogram are equal to two adjacent sides and an angle of another parallelogram, the parallelograms are equal in all respects. 21. If the diagonals and the angle between them of one parallelogram are equal respectively to the diagonals and the angle between them of another parallelogram, the parallelograms are equal in all respects. 22. Lines drawn parallel to one pair of sides and terminated by the other pair of sides of a parallelogram are equal. 23. The middle points of the series of lines of Question 22 all lie on a straight line. 24. Lines drawn through the middle point of one side of a triangle parallel to the other two sides bisect those sides. 25. The line joining the middle points of two sides of a triangle is parallel to the third side. 26. The lines joining the middle points of the sides of a triangle divide the triangle into four triangles which are equal in all respects. 27. The four lines joining the middle points of pairs of adjacent sides of a quadrilateral form a parallelogram. 28. The two lines joining the middle point of opposite sides of a quadrilateral bisect each other. 29. ABDC is a parallelogram; A is joined to the middle point E of DC and D is joined to the middle point F of AB, prove that AE and DF are parallel. 30. Prove that in the figure of Question 29 BC is trisected. 31. Lines drawn parallel to a given straight line and terminated by the points in which they cut two given parallel straight lines, are equal. 32. Lines drawn from points in one of two given parallel straight lines perpendicular to the other are equal. 33. A series of lines of given length are drawn from points in a given straight line, making a given angle with it; prove that their extremities all lie in a line parallel to the given straight line. SECTION VIII. EQUALITY OF AREAS. 132. DEF. Two parallelograms are said to be between the same parallels when a pair of opposite sides of one parallelogram are in the same straight lines respectively with a pair of opposite sides of the other parallelogram. Thus ABDC and EFGH are between the same parallels if AD, EH are in one straight line and BC, FG are in one other straight line. 133. DEF. Two triangles are said to be between the same parallels when a side of one is in the same straight line with a side of the other and the line joining their opposite angular points is parallel to that straight line. Thus ABC, DEF are between the same parallels if BC, EF are in a straight line, and AD is parallel to BCEF. 134. DEF. The altitude of a parallelogram with reference to a pair of its sides is the perpendicular distance between those sides. Thus, in the first figure on p. 126, if KN be perpendicular to FG, then KN is the altitude of EFGH with reference to the base FG. 135. DEF. The altitude of a triangle with reference to a given side is the length of the perpendicular drawn from the opposite angular point to that side. Thus, in the second figure on p. 126, if AN is perpendicular to BC, then AN is the altitude of ABC, with reference to the base BC. EXAMPLES XL. 1. Prove that parallelograms which are between the same parallels are of equal altitudes. 2. Prove that parallelograms of the same altitude can be placed so as to be between the same parallels. 3. Prove that, if two triangles are such that the perpendicular drawn from an angular point to the opposite side in one triangle, is equal to a corresponding line in the other the triangles can be placed so as to be between the same parallels. 4. Two triangles are between the same parallels and the angles which the sides of one triangle make with the parallels are equal to the angles which the sides of the other make with the parallels; prove that the triangles are equal in all respects. |