q.e.d. OPD=OBD, which is not true, . . AB and EF cannot be unequal, or AB=EF. 3. If OBD=OFH and AB=EF, then AD=EH. Proof.—Let AB and EF be taken as bases and AD and EH as altitudes (138°), and the theorem follows from the second part. 9.e.d. Cor. In any rectangle we have the three parts, base, altitude, and area. If any two of these are given the third is given also. E B F D 140°. Theorem.-A parallelogram is equal to the rectangle ç on its base and altitude. AC is a Owhereof AD is the base and DF is the altitude. Then OAC= on AD and DF. Proof.—Complete the CADFE by drawing AE I to CB produced Then AAEB=ADFC, :;: AE=DF, AB=DC, and LEAB=LFDC; .: ADFC may be transferred to the position AEB, and OABCD becomes the DAEFD, .. JAC=O on AD and DF. q.e.d. Cor. 1. Parallelograms with equal bases and equal altitudes are equal. For they are equal to the same rectangle. Cor. 2. Equal parallelograms with equal bases have equal altitudes, and equal parallelograms with equal altitudes have equal bases. Cor. 3. If equal parallelograms be upon the same side of the same base, their sides opposite the common base are in line. 141°. Theorem.-A triangle is equal to one-half the rectangle on its base and altitude. B E с ABC is a triangle of which AC is the base and BE the altitude. Then AABC=o on AC and BE. Proof.—Complete the OABDC, of which AB and AC are adjacent sides. Á Then AABC = ADCB, AABC=OAD=} on AC and BE. (140) q.e.d. Cor. I. A triangle is equal to one-half the parallelogram having the same base and altitude. Cor. 2. Triangles with equal bases and equal altitudes are equal. For they are equal to one-half of the same rectangle. Cor. 3. A median of a triangle bisects the area. For the median bisects the base. Cor. 4. Equal triangles with equal bases have equal altitudes, and equal triangles with equal altitudes have equal bases. Cor. 5. If equal triangles be upon the same side of the same base, the line through their vertices is parallel to their common base. B 142°. Theorem.-If two triangles are upon opposite sides of the same base 1. When the triangles are equal, the base bisects the segment joining their vertices; 2. When the base bisects the segment joining their vertices, the triangles are equal. (Converse of 1.) ABC and ADC are two triangles upon opposite sides of the common base AC. AABC=AADC, then BH=HD. q.e.d. 2. If BH=HD, then AABC=AADC. H с 1. If E Proof.—Since BH=HD, .AABH=AADH, and ACBH=ACDH. (141°, Cor. 3) .. adding, AABC=AADC. 9.e.d. 143. Def.—By the sum or difference of two closed figures is meant the sum or difference of the areas of the figures. If a rectangle be equal to the sum of two other rectangles its area may be so superimposed upon the others as to cover both. B D F Х Y А с 144o. Theorem.-If two rectangles have equal altitudes, their sum is equal to the rectangle on their common altitude and the sum of their bases. Proof.—Let the Os X and Y, having equal altitudes, be so placed as to have Ě their altitudes in common at CD, and so that one may not overlap the other. Then LBDC=LCDF= 7, ... BDF is a line. (38°, Cor. 2) Similarly ACE is a line. But BD is || to AC, and BA is || to DC || to FE; therefore AF is the o on the altitude AB and the sum of the bases AC and CE; and the DAF=DAD +CF. q.e.d. Cor. 1. If two triangles have equal altitudes, their sum is equal to the triangle having the same altitude and having a base equal to the sum of the bases of the two triangles. Cor. 2. If two triangles have equal altitudes, their sum is equal to one-half the rectangle on their common altitude and the sum of their bases. Cor. 3. If any number of triangles have equal altitudes, their sum is equal to one-half the rectangle on their common altitude and the sum of their bases. In any of the above, “base” and “altitude” are interchangeable. 145o. Theorem.—Two lines parallel to the sides of a parallelogram and intersecting upon a diagonal divide the parallelogram into four parallelograms such that the two through which the diagonal does not pass are equal to one another. In the OABCD, EF is || to AD and GH is || to BA, and these intersect at O on the diagonal AC. Then BO=OOD. B G с E to A HD Proof.—AABC=AADC, and AAEO=AAHO, and AOGC=A0FC; (141°, Cor. 1) but JBO=AABC-AAEO-AOGC, and DOD=AADC-AAHO-AOFC. OBO=OOD. q.e.d. Cor. 2. If OBO=OOD, O is on the diagonal AC. (Converse of the theorem.) For if () is not on the diagonal, let the diagonal cut EF in O'. Then OBO'= DO'D. (145) But OBO’ is <OBO, and DO'D is > DOD; .. BO is > OOD, which is contrary to the hypothesis; .. the diagonal cuts EF in O. B A D E Ex. Let ABCD be a trapezoid. (84°, Def.) In line with AD make DE=BC, and in line with BC make CF=AD. Then BF=AE and BFEA is a But the trapezoid CE can be superimposed on the trapezoid DB, since the sides are respectively equal, and LF=A, and LE=B, etc. trapezoid BD="OBE, or, a trapezoid is equal to one-half the rectangle on its altitude and the sum of its bases. G EXERCISES. 1. To construct a triangle equal to a given quadrangle. 2. To construct a triangle equal to a given polygon. 3. To bisect a triangle by a line drawn through a given point in one of the sides. 4. To construct a rhombus equal to a given parallelogram, and with one of the sides of the parallelogram as its side. 5. The three connectors of the middle points of the sides of a triangle divide the triangle into four equal triangles. 6. Any line concurrent with the diagonals of a parallelogram bisects the parallelogram. 7. The triangle having one of the non-parallel sides of a trapezoid as base and the middle point of the opposite side as vertex is one-half the trapezoid. 8. The connector of the middle points of the diagonals of a quadrangle is concurrent with the connectors of the middle points of opposite sides. 9. ABCD is a parallelogram and O is a point within. Then AAOB+ACOD=o. What does this become when 0 is without ? 10. ABCD is a parallelogram and O is a point within. Then AAOC=AAOD-AAOB. theorem is important in the theory of Statics.) 1. Bisect a trapezoid by a line through the middle point of one of the parallel sides. By a line through the middle point of one of the non-parallel sides. 12. The triangle having the three medians of another tri angle as its sides has three-fourths the area of the other. |