Proof The sum of the angles of a quadrilateral is equal to 4 rt. ▲ s, (ii) A quadrilateral is a parallelogram if both pairs of opposite sides are equal. (Draw a diagonal and prove the two triangles congruent.) (iii) A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel. (Draw a diagonal and prove the two triangles congruent.) (iv) A quadrilateral is a parallelogram if its diagonals bisect one another. (Prove two opposite triangles congruent.) COR. If equal perpendiculars are erected on the same side of a straight line, the straight line joining their extremities is parallel to the given line. THE MID-POINT THEOREMS. THEOREM 22. The straight line drawn through the middle point of one side of a triangle parallel to another side bisects the third side. A B F Fig. 42. E THEOREM 23. The straight line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of it. Data ABC is a triangle. D, E are the mid-points of AB, AC To prove that DE is || to BC, and A E Now CF is equal and || to DB, .. DBCF is a ||ogram, Th. 21 (iii). ..DE=DF = BC (DF, BC are opp. sides of ||ogram DBCF). Q. E. D. INTERCEPTS BY PARALLEL LINES. THEOREM 24. If there are three or more parallel straight lines, and the intercepts made by them on any one straight line that cuts them are equal, then the corresponding intercepts on any other straight line that cuts them are also equal. Data The parallels AB, CD, EF are cut by the straight lines ACE, BDF, and the intercepts AC, CE are equal. To prove that the corresponding intercepts BD, DF are equal. Construction Through B draw BH || to ACE to meet CD at H. Through D draw DK || to ACE to meet EF at K. Proof H K Fig. 44. CONSTRUCTION. To divide a given straight line into any number of equal parts (say five). Let AB be the given straight line. Construction Through A draw AC making any angle with AB. Proof From AC cut off any part AD. From DC cut off parts DE, EF, FG, GH, equal to AD, so that AH is five times AD. For Exercises see Practical Geometry, p. 146. PARALLELOGRAM. Ex. 1. If the diagonals of a parallelogram are equal, it must be a rectangle. Ex. 2. Prove that, if all the sides of a quadrilateral are equal, the figure is a parallelogram and its diagonals cut at right angles. Ex. 3. An equilateral four-sided figure with one of its jangles a right angle must be a square. Ex. 4. Two straight lines bisect one another at right angles; prove that they are the diagonals of a rhombus. Ex. 5. If the diagonals of a quadrilateral are equal and bisect one another at right angles, the quadrilateral must be a square. Ex. 6. The straight line joining the mid-points of two opposite sides of a parallelogram is parallel to the other two sides. Ex. 7. ABCD is a parallelogram (not rectangular), and AL and CM are the perpendiculars from A and C on to the diagonal BD. Prove (otherwise than by symmetry) that ALCM is a parallelogram. Ex. 8. In the diagonal AC of a parallelogram ABCD points P, Q are taken such that APCQ; prove that BPDQ is a parallelogram. Ex. 9. E, F, G, H are points in the sides AB, BC, CD, DA respectively of a parallelogram ABCD, such that AH=CF and AE=CG: show that EFGH is a parallelogram. Ex. 10. The figure formed by joining the mid-points of the sides of a rectangle is a rhombus. Ex. 11. ABCD is a square; from A lines are drawn to the mid-points of BC, CD; from C lines are drawn to the mid-points of DA, AB. Prove that these lines enclose a rhombus. Ex. 12. On the sides of a parallelogram ABCD and outside the parallelogram equilateral triangles are described whose other vertices are E, F, G, H. Prove that EFGH is a parallelogram. Ex. 18. Any straight line drawn through O, the point of intersection of the diagonals of a parallelogram, and terminated by the sides of the parallelogram must be bisected at O. Ex. 14. The bisectors of two adjacent angles of a parallelogram are at right angles to one another. Ex. 15. The bisectors of two opposite angles of a parallelogram are parallel. Ex. 16. ABCD is a parallelogram; AB, CD are bisected at X, Y respectively; prove that BXDY is a parallelogram. Ex. 17. PQRS is a parallelogram; X is the mid-point of PQ; RX and SP are produced to meet at Y. Prove that PYPS. Ex. 18. The diagonal AC of a parallelogram ABCD is produced to E, so that CE=CA; through E, EF is drawn parallel to CB to meet DC produced in F. Prove that ABFC is a parallelogram. Ex. 19. The diagonal AC of parallelogram ABCD is produced to P, so that CP CA; through P and B, PQ, BQ are drawn parallel to CB, AC respectively. Prove that ABQC is a parallelogram. Ex. 20. ABCD is a parallelogram; the side AB is produced to E so that BE=BC; and BA to F so that AF-AD; show that EC and FD produced meet at right angles. Ex. 21. ABCD is a parallelogram and AD=2AB; AB is produced both ways to E, F so that EA AB=BF. Prove that CE, DF intersect at right angles. S. H. T. G. 4 |