Because AB is parallel to CD, and BC meets them, the alternate angles ABC, BCD are equal (21. 1.) to one another and because AC is parallel to BD, and BC meets them, the alternate angles ACB, CBD are equal (21. 1.) to one another; wherefore the two triangles ABC, CBD have two angles ABC, BCA in one, equal

to two angles BCD, CBD in the other, each to each, and the side BC, which is adjacent to these equal angles, common to the two triangles; therefore their other sides are equal, each to each, and the third angle of the one to the third angle of the other (Th. 2.); viz. the side AB to the side CD, and AC to BD, and the angle BAC equal to the angle BDC. And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, the whole angle ABD is equal to the whole angle ACD: And the angle BAC has been shewn to be equal to the angle BDC: therefore the opposite sides and angles of a parallelogram are equal to one another; also, its diameter bisects it; for AB being equal to CD, and BC common, the two AB, BC are equal to the two DC, CB, each to each; now the angle ABC is equal to the angle BCD; therefore the triangle ABC is equal (1. 1.) to the triangle BCD, and the diameter BC divides the parallelogram ACDB into two equal parts.

COR. 1. Two parallel lines, included between two other parallels, are equal.

COR. 2. If one angle of a parallelogram be a right angle, all the other three will also be right angles, and the parallelogram will be a rectangle. COR. 3. Hence, two parallels are every where equally distant.

COR. 4. Hence, also, the sum of any two adjacent angles of a parallelogram is equal to two right angles.

PROP. XXIX. THEOR.

Parallelograms upon the same base and between the same parallels, are equivalent to one another.

(SEE THE 2d and 3d figures.)

Let the parallelograms ABCD, EBCF be upon the same base BC, and between the same parallels AF, BC; the parallelogram ABCD is equal to the parallelogram EBCF.

If the sides AD, DF of the parallelograms ABCD, DBCF opposite to the base BC be terminated in the same point D; it is plain that each of the parallelograms is double (28. 1.) of the triangle BDC; and they are therefore equal to one another.

A

B

D

C

F

But, if the sides AD, EF, opposite to the base BC of the parallelogramis ABCD, EBCF, be not terminated in the same point; then, because ABCD is a parallelogram, AD is equal (28. 1.) to BC; for the same reason EF is

equal to BC; wherefore AD is equal (1. Ax.) to EF; and DE is common; therefore the whole, or the remainder, AE is equal (2. or 3. Ax.) to the whole, or the remainder DF; now AB is also equal to DC; therefore the two EA, AB are equal to the two FD, DC, each to each; but the ex

terior angle FDC is equal (21. 1.) to the interior EAB, wherefore the base EB is equal to the base FC, and the triangle EAB (1. 1.) to the triangle FDC. Take the triangle FDC from the trapezium ABCF, and from the same trapezium take the triangle EAB; the remainders will then be equal (3. Ax.), that is, the parallelogram ABCD is equal to the parallelogram EBCF.

PROP. XXX. THEOR.

Parallelograms upon equal bases, and between the same parallels, are equivalent to one another.

Let ABCD, EFGH be parallelograms upon equal bases BC, FG, and between the same parallels AH, BG; the parallelogram ABCD is equal to EFGH.

A

B

DE

C

F

H

Ꮐ

Join BE, CH; and because BC is equal to FG, and FG to (28. 1.) EH, BC is equal to EH ; and they are parallels, and joined towards the same parts by the straight lines BE, CH: But straight lines which join equal and parallel straight lines towards the same parts, are themselves equal and parallel (27. 1.); therefore, EB, CH are both equal and parallel, and EBCH is a parallelogram; and it is equal (29. 1.) to ABCD, because it is upon the same base BC, and between the same parallels BC, AH; For the like reason, the parallelogram EFGH is equal to the same EBCH: Therefore also the parallelogram ABCD is equal to EFGH.

PROP. XXXI. THEOR.

Triangles upon the same base, and between the same parallels, are equivalent to one another.

Let the triangles ABC, DBC be upon the same base BC, and between the same parallels, AD, BC: The triangle ABC is equal to the triangle DBC.

Produce AD both ways to the points E, F, and through B draw BE parallel to CA; and through C draw CF parallel to BD: Therefore, each of the figures EBCA, DBCF is a parallelogram; and EBCA is equal (29. 1.) to DBCF, because they are upon the same base BC, and between the same pa

E

A D

F

B

rallels BC, EF; but the triangle ABC is the half of the parallelogram EBCA, because the diameter AB bisects (28. 1.) it; and the triangle DBC is the half of the parallelogram DBCF, because the diameter DC bisects it; and the halves of equal things are equal (7. Ax.); therefore the triangle ABC is equal to the triangle DBC.

PROP. XXXII. THEOR.

Triangles upon equal bases, and between the same parallels, are equivalent to one another.

Let the triangles ABC, DEF be upon equal bases BC, EF, and between the same parallels BF, AD: The triangle ABC is equal to the triangle DEF.

G

A

D

H

Produce AD both ways to the points G, H, and through B draw BG parallel to CA, and through F draw FH parallel to ED: Then each of the figures GBCA, DEFH is a parallelogram; and they are equal to (30. 1.) one another, because they are upon equal bases BC, EF, and between the same parallels BF, GH; and the triangle ABC is the half (28. 1.) of the parallelogram GBCA, because the diameter AB bisects it; and the triangle DEF is the half (28. 1.) of the parallelogram DEFH, because the diameter DF bisects it; But the halves of equal things are equal (7. Ax.); therefore the triangle ABC is equal to the triangle DEF.

B

CE

F

Equivalent triangles upon the same base, and upon the same side of it, are between the same parallels.

Let the equal triangles ABC, DBC be upon the same base BC, and upon the same side of it; they are between the same parallels.

Join AD: AD is parallel to BC; for, if it is not, through the point A draw AE parallel to BC, and join EC: The triangle ABC, is equal (31. 1.)

to the triangle EBC, because it is upon the same base BC, and between the same parallels BC, AE; But the triangle ABC is equal to the triangle BDC; therefore also the triangle BDC is equal to the triangle EBC, the greater to the less, which is impossible: Therefore AE is not parallel to BC. In the same manner, it may be demonstrated that no other line but AD is parallel to BC; AD is therefore parallel to it.

PROP. XXXIV. THEOR.

B

D

E

Equivalent triangles on the same side of bases which are equal and in the same straight line, are between the same parallels.

Let the equal triangles ABC, DEF be upon equal bases BC, EF, in the same straight line BF, and to

wards the same parts; they are between the same parallels.

B

A

D

C E

F

Join AD; AD is parallel to BC; For, if it is not, through A draw AG parallel to BF, and join GF. The triangle ABC is equal (32. 1.) to the triangle GEF, because they are upon equal bases BC, EF, and between the same parallels BF, AG; But the triangle ABC is equal to the triangle DEF; therefore also the triangle DEF is equal to the triangle GEF, the greater to the less, which is impossible; Therefore AG is not parallel to BF; and in the same manner it may be demonstrated that there is no other parallel to it but AD; AD is therefore parallel to BF.

PROP. XXXV. THEOR.

If a parallelogram and a triangle be upon the same base, and between the same parallels; the parallelogram is double of the triangle.

Let the parallelogram ABCD and the triangle EBC be upon the same base BC and between the same parallels BC, AE; the parallelogram ABCD is double of the triangle EBC.

Join AC; then the triangle ABC 'is equal (31. 1.) to the triangle EBC, because they are upon the same base BC, and between the same parallels BC, AE. But the parallelogram ABCD is double (28. 1.) of the triangle ABC, because the diameter AC divides

A

B

7

D

E

it into two equal parts; wherefore ABCD is also double of the triangle

EBC.

PROP. XXXVI. THEOR.

The complements of the parallelograms which are about the diameter of any parallelogram, are equivalent to one another.

Let ABCD be a parallelogram of which the diameter is AC; let EH, FG be the parallelograms about AC, that is, through which AC passes, and let BK, KD be the other parallelograms, which make up the whole figure ABCD, and are therefore called the complements; The complement BK is equal to the complement KD.

A

H

D

E

K

F

B

Ꮐ

C

Because ABCD is a parallelogram and AC its diameter, the triangle ABC is equal (28. 1.) to the triangle ADC; And because EKHA is a parallelogram, and AK its diameter, the triangle AEK is equal to the triangle AHK: For the same reason, the triangle KGC is equal to the triangle KFC. Then, because the triangle AEK is equal to the triangle AHK, and the triangle KGC to the triangle KFC; the triangle AEK, together with the triangle KGC, is equal to the triangle AHK, together with the triangle KFC: But the whole triangle ABC is equal to the whole ADC; therefore the remaining complement BK is equal to the remaining complement KD.

PROP. XXXVII. THEOR.

In any right angled triangle, the square which is described upon the side subtending the right angle, is equivalent to the squares described upon the sides which contain the right angle.

Let ABC be a right angled triangle having the right angle BAC; the square described upon the side BC is equal to the squares described upon BA, AC.

On BC describe the square BDEC, and on BA, AC the squares GB, HC; and through A draw AL parallel to BD or CE, and join AD, FC; then, because each of the angles BAC, BAG is a right angle (24. def.), the two straight lines AC, AG upon the opposite sides of AB, make with it at the point A 'he adjacent angles equal to two right angles; therefore CA is in the same straight line (7. 1.) with AG; for the same reason, AB and AH are in the same straight line.