each other [vii]: but the triangle A B C is half the parallelogram AH, and the triangle D E F is half the parallelogram FI [III. 28i]; therefore the triangles A B C and D E F are equivalent to each other [i], W. W. T. B. D.

COROLLARY I. If two triangles having the same base or equal bases, be between the same parallels, they are equivalent to each other. (Eucl. I. 37 and 38.)

COROLLARY II. If two triangles have two sides of the one equal to two sides of the other, each to each, and if the angles formed by these sides be supplementary, these triangles are equivalent to each other.

COROLLARY III. If the points which divide the base of a triangle into equal parts be joined to the apex, the partial triangles thus formed are equivalent to each other.

COROLLAKY IV. The diagonals of a parallelogram divide it into four equivalent triangles, and conversely.

COROLLARY V. Of the four triangles formed by the sides and diagonals of a trapezium, those formed by the lateral sides are equivalent to each other, and conversely.

Scholium. The perimeters of two equivalent triangles having equal bases may differ infinitely.

THEOREM 5.

If two equivalent triangles have equal bases, they have equal altitudes.

Let there be two equivalent triangles, A B C and A D C, having equal bases.

Let these triangles be applied to each other so that the bases coincide with each other in A C; through the apex D, let there be a parallel to A C, cutting CB or its prolongation in some point E; and let A E be joined. Because the triangles A E C and ADC are between the parallels AC and E D, they are equivalent to each other [31]; because ABC is equivalent to ADC [Hyp.], the triangles A B C and AEC are equivalent to each other [i]; therefore the triangle ABE has no magnitude: that is, B and E are one and the same point. Be

cause the vertices B and D are on the same parallel to A C, they are

E

B

C

equally distant from AC [I. 19]; therefore the two triangles A B C and A D C have equal altitudes, W. W. T. B. D. Inversely

If two equivalent triangles have unequal altitudes, they have unequal bases.

COROLLARY I. If two equivalent triangles have the same base or equal bases, on a straight line and on the same side thereof, they are between the same parallels. (Eucl. I. 39 and 40.)

COROLLARY II. If two equivalent triangles have equal altitudes, they have equal bases.

THEOREM 6.

(Eucl. I. 43.)

The complements of two diagonal parallelograms are equivalent to each other.

Let A EKH and KICF be two diagonal parallelograms, and let HBFK and EDIK be their complements.

Let A KC be the diagonal of the parallelogram DB; then AK is the diagonal of the parallelogram E H, and KC that of the parallelogram IF [III. Def. 21]. Because the triangle AHK is equivalent to the triangle A E K [III. 29 i], and the triangle KFC is equivalent to the triangle KI C, the triangles AHK and K F C are together equi

A

E

valent to the triangles A E K and K IC [ii]: but the triangle ABC is equivalent to the triangle A D C ; therefore the difference between

H

B

F

D

I

the triangle ABC and the sum of the triangles A H K and K F C is equivalent to the difference between the triangle A D C and the sum of the triangles A E K and K I C [iv]; that is, the parallelograms K B and K D are equivalent to each other, W. W. T. B. D.

COROLLARY. If two straight lines be drawn parallel to the sides of a parallelogram through any point of a diagonal, the partial parallelograms determined by one of them, are equivalent to those determined by the other.

SECTION II.

ON EQUIVALENT RIGHT-ANGLED QUADRANGLES.

DEFINITIONS.

3. The square of a straight line is the square having that line for one of its sides.

It has been shown [III. 39 i] that the squares of equal straight lines are not only equivalent, but equal to each other.

4. The rectangle of two straight lines, or of one straight

line by another, is the rectangle having these lines for its adjacent sides.

The sides adjacent to the base of a rectangle are perpendicular to that base [III. Def. 19]; hence, two rectangles having equal bases and altitudes are not only equivalent, but equal to each other [III. 39 ii]. Also, if two straight lines be equal to two others, each to each, the rectangle of the first two is equal to that of the others.

THEOREM 7.

(Eucl. II. 1.)

The rectangle of a straight line by the sum of several straight lines, is equivalent to the sum of the rectangles of the first line by each of the others.

Let a straight line AB be the sum of the straight lines A C, C D, DE and EB; let there be another straight line B F, and let ABFH be the rectangle of A B and B F.

Let CI, DK and EL be perpendiculars to AB; then AI, C K DL and E F are rectangles [III. Def. 19], and E F is the rectangle of EB by BF [Def. 4.]. Because CI, DK, EL are each equal to BF [I. 19], the rectangles AI, CK, D L are respectively equal to the rectangles of AC, CD, DE by BF; therefore the rectangle AF is equivalent to the sum of the rectangles of A C, CD, DE and EB by BF, which W. T. B. D.

COROLLARY I. The square of the sum of several straight lines, is equivalent to the sum of its rectangles by each line. (Eucl. II. 2.)

COROLLARY II. The rectangle of the sum of several straight lines by one of them, is equivalent to the square of the latter together with its rectangle by the sum of the others. (Eucl. II. 3.)

COROLLARY III. The rectangle of

the sum of several straight lines by the sum of several other straight lines, is equivalent to the sum of the rectangles of each line of the first sum by each line of the second.

THEOREM 8.

(Eucl. II. 4.)

The square of the sum of two straight lines, is equivalent to the sum of their squares together with twice their rectangle.

K

AC and CB, and let A B D E be the

square of A B.

Let CH be a perpendicular to AB; let A I be equal to AC and let IF be a perpendicular to AE, cutting CH in a point K: then AK is the square of AC [Def. 3], and CF is equivalent to the rectangle of

AC by CB. Because IE and CB are equal to each other [iv], HF is equivalent to the square of CB, and I H is equivalent to the rectangle of AC by CB; therefore the square A D is equivalent to the