COROLLARY III. Two parallelograms which have one equal angle formed by proportional sides are similar, to each other.

COROLLARY IV. Two trapeziums which have proportional sides are similar to each other.

THEOREM 43.

If two polygons have the angles formed by one side of the one with the adjacent sides and diagonals equal, each to each, to the angles formed by one side of the other with the adjacent sides and diagonals, they are similar to each other.

Let the angles formed by the side AH of the polygon ABCDEFGH with the adjacent sides and diagonals, be equal each to each to the angles formed by the side A'H' of the polygon A'B'C' D'E' F'G' H' with the adjacent sides and diagonals.

B

D

G

Α'

If AH be equal to A'H', then the two polygons are equal [40],

and consequently similar, to each other.

If AH be not equal to A'H', then, because the triangles A B H and A'B' H' have the angles BAH and BH A respectively equal to the angles B'A'H' and B'H' A' [Hyp.], they are similar to each other [23]; therefore A B is to A'B' as A H is to A'H' [22].

Because the triangles ACH and A' C'H' have the angles CAH and CHA respectively equal to the angles C'A'H' and C'H' A' [Hyp.], they are similar to each other [23]; hence AC is to A'C' as AH is to A'H' [22]: but already A B is to A'B' as A H is to A'H' [Dem.]; therefore AB is to A'B' as AC is to A'C' [i]. Now, the triangles ABC and A'B'C' have the angles BAC and B' A'C' equal to each other [Hup.], and formed by proportional sides [Dem.]; therefore, they are similar to each other [24].

Because the triangles ADH and A' D'H' have the angles DAH and DHA respectively equal to the angles D' A'H' and D'H' A'[Hyp.], they are similar to each other [23]; hence A D is to A'D' as A His to A'H' [22]; but A C is to A'C' as AH is to A'H' [Dem.]; therefore AC is to A'C' as AD is to A' D' [i]. Now, the triangles A CD and A'C' D' have the angles CAD and C'A' D' equal to each other [Hyp.], and formed by proportional sides [Dem.]; therefore they are similar to each other [24].

The same reasoning would prove that the remaining triangles of the polygon ABCDEFGH are similar, each to each, to the remaining triangles of the polygon A' B'C' D'E' F'G'H'; that is, the two polygons are composed of the same number of triangles similar each to each and similarly placed: therefore they are similar to each other [42], W. W. T. B. D.

THEOREM 44.

Two parallelograms about the same straight line are similar to each other.

Let there be two parallelograms ABCD and CEF H, having the

vertex C in common; let the sides AD and AB be parallel to the sides C H and CE respectively, and let the diagonals AC and CF be the continuation of each other.

B

F

H

Because CB and CH are each parallel to DA [Hyp.], they are parallel to each other [I. 17], and C H and C B are the continuation of each other [I. 17 iv]: likewise CD and CE are the continuation of each other; therefore the angles BCD and ECH are equal to each other [II. 2].

Because the triangles A D C and CHF have the sides of the one parallel to the sides of the other, they are similar to each other [23 i]; therefore A D is to CH as DC is to FH [22]: but C B and CE are respectively equal to A D and F H [29]; then C Bis to C Has CD is to CE; therefore the parallelograms B D and E H are similar to each other [42 iii], W. W. T. B. D.

COROLLARY. If a parallelogram be divided into four partial parallelograms by parallels to two adjacent sides through a point of one diagonal, the two partial parallelograms which are about the diagonal, are similar to the whole.

THEOREM 45.

If two similar parallelograms similarly placed have an angle in common, they are about the same straight line.

Let A B C D and B E F H be two similar parallelograms similarly placed and having the angle A B C in common.

Let B D be a diagonal of the parallelogram A C and let I be the point where BD cuts EF; let IK be parallel to EB: then the parallelograms K E and AC are about the line BD [Def. 21]. Because the parallelograms K E and AC are similar to each other [44], BC is to BE as BA is to BK [41 ii]. But the parallelograms HE and A C are also similar to each other [Hyp.], then B C is to BE as BA is to BH: therefore B K is equal to B H [xvii]; that is, H and K are one and the same point. Because I K and F H are parallel to EB, they are parallel to each other [I. 17 iii], and coincide with each other [I. 17 iv]: that is, the parallelograms K E and HE are one and the same figure; therefore the parallelograms HE and AC are about the line BD [Def. 21], which W. T. B. D.

COROLLARY. If two similar parallelograms similarly placed have vertical angles, they are about the same straight line.

CHAPTER II.

ON SYMMETRIC PLANE FIGURES.

SECTION I.

ON SYMMETRIC POLYGONS.

DEFINITIONS.

28. Two plane figures are symmetric, or placed symmetri cally, to an axis when the extremity of every perpendicular drawn from the perimeter of one of them, to a straight line called their line, or axis, of symmetry, and produced beyond it to a distance equal to itself, is a point of the perimeter of the other; each of the two figures is the symmetric of the other; two points of their perimeters are termed symmetric when the line of symmetry is the middle perpendicular on the straight line joining them.

29. A plane figure is symmetric to an axis when all points of its perimeter on the same perpendicular to an interior straight line, called axis of symmetry, or simply axis, are equally distant from that line: two points of the perimeter which are on the same perpendicular to the axis are termed symmetric.

It follows from this definition that if one of the sides of a polygon symmetric to an axis be met by the axis, the latter is the middle perpendi cular on that side.