but all the interior angles, together with four right angles, are equal to twice as many right angles as the figure has sides; (Th. 26.) therefore all the exterior angles are equal to four right angles*. : Q. E. D. THEOREM 27. The adjoining angles of a parallelogram are supplementary and the opposite angles are equal. Part. En. Let HBGE be a parallelogram; that is, let HE, EG be respectively parallel to BG, BH; (Def. 37.) it is required to prove that its adjoining angles EHB, HBG are supplementary, and its opposite angles HBG, HEG are equal. Proof. Because HE is parallel to BG, (Hyp.) and HB meets them, therefore HBG is supplementary to EHB. (Th. 23. Cor.) And because HB is parallel to EG, (Hyp.) and HE meets them, therefore HEG is supplementary to EHB; (Th. 23. Cor.) but HBG is also supplementary to EHB, therefore HEG is equal to HBG. (Th. 1. Cor. 3.) Q. E. D. COR. I. Hence if one of the angles of a parallelogram is a right angle, all its angles are right angles. * Euclid, 1. 32. Cor. 2. Cor. 2. If two straight lines are respectively parallel to two other straight lines they will include equal angles towards the same parts. Def. 40. A right-angled parallelogram is called a rectangle. THEOREM 28. The opposite sides of a parallelogram are equal to one another, and the diagonal bisects it. Part. En. Let ABCD be a parallelogram, that is, let AB be parallel to CD, and AD to BC; it is required to prove that AB ist equal to DC, and AD to BC. Proof. Join AC. Then because AB is parallel to DC, and AC meets them; (Hyp.) therefore the angle BAC is equal to the alternate angle ACD. (Th. 22.) And because AD is parallel to BC; (Hyp.) therefore the angle BCA is equal to the alternate angle CAD: (Th. 22.) therefore in the triangles BAC, DCA we have the angle BAC=the angle DCA, 1 and the angle BCA = the angle DAC; and the side AC adjacent to the equal angles common; therefore the triangles are equal in all respects, (Th. 7.) that is, AB is equal to DC, AD to BC, and the area ABC to the area ADC* Q. E. D. COR. Hence if the adjacent sides of a parallelogram are equal, all its sides are equal. Def. 41. A parallelogram all whose sides are equal is called a rhombus. Def. 42. A square is a rectangle that has all its sides equal. THEOREM 29. If two parallelograms have two adjacent sides of the one respectively equal to two adjacent sides of the other, and likewise an angle of the one equal to an angle of the other; the parallelograms are identically equal. Part. En. Let ABCD, EFGH be two parallelograms which have two adjoining sides AB, BC of the one equal respectively to two adjoining sides EF, FG of the other, and have likewise the included angles B and F equal ; it is required to prove that the parallelograms are identically equal. Proof. For if the point B were placed on F, and the line BC along the line FG; then because BC = FG, (Hyp.) therefore the point C will fall on G; and because the angle ABC = the angle EFG, (Hyp.) * Euclid, I. 34. therefore BA will fall along FE; and because BA= FE, (Hyp.) therefore the point A will fall on E. And because AD is parallel to BC, (Hyp.) therefore AD will fall along EH; (Ax. 5.) and similarly CD will fall along GH, and therefore the point D will fall on the point H; that is, the parallelograms are identically equal. Q. E. D. Cor. Two rectangles are equal, if two adjacent sides of the one are respectively equal to two adjacent sides of the other; and two squares are equal, if a side of the one is equal to a side of the other. THEOREM 30. . If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram. Part. En. Let ABCD be a quadrilateral in which the opposite sides AB, CD are equal and parallel ; it is required to prove that AC is equal and parallel to BD. Proof. Join AD. (Hyp.) therefore the angle BAD is equal to the alternate angle ADC; (Th. 22.) and therefore in the triangles BAD, CDA, we have BA = CD, (Hyp.) ) AD common, equal; therefore the triangles are equal in all respects; (Th. 5.) that is, BD= AC; and the angle BDA = the angle CAD; but these are alternate angles; and therefore AC is parallel to BD* (Th. 21.) Q. E. D. THEOREM 31. Straight lines which are equal and parallel have equal projections on any other straight line; conversely, parallel straight lines which have equal projections on another straight line are equal; and equal straight lines, which have equal projections on another straight line, are equally inclined to that line. a b c d Part. En. Let AB, CD be equal and parallel straight lines, and let ab, cd be their projections on any other straight line. Then shall ab be equal to cd.. Proof. Through A, C draw AE, CF parallel to abcd, meeting Bb, De in E, F. * Euclid, I. 33. |