Conversely, equal triangles upon the same base or upon equal bases in the same straight line are between the same parallels.

MEASUREMENT OF FIGURES.

DEF. Every right-angled parallelogram is called a rectangle.

The altitude of a parallelogram is the perpendicular distance of the opposite side from the base.

The altitude of a triangle is the perpendicular distance of the vertex from the base.

THEOREM V.

The measure of a rectangle is the product of its base multiplied by its altitude, the square on the unit of length being taken as the unit of surface.

Divide AB, AD into parts each equal to the unit of length, and through the points of division draw straight lines parallel to AB and AD.

the

The figures so formed are all squares, and all equal to square KH.

[III. 2.

Hence if AB contain b equal parts, KB will contain b squares each equal to KH.

Therefore KB = KHxb.

Again the figures DG, FE are equal to each other and

to KB.

[III. 2.

Hence if AD contain h equal parts, DB will contain h figures each equal to KB.

Therefore DB = KB × h,

but KB-KH × b.

Therefore DB = KH xbx h.

Hence if KH be taken for the unit of surface,

the measure of DB = b × h,

where b and h are the numbers which express the lengths of

The measure of a parallelogram is the product of its base multiplied by its altitude.

Let ABCD be a parallelogram, draw BE perpendicular to AB, and complete the parallelogram ABEF.

Then the parallelogram ABCD = the rectangle ABEF. Therefore its measure = AB × BE.

COR. 1. Parallelograms of the same or equal altitude are to one another as their bases.

Let ABCD, EFGH be parallelograms of equal altitude.

COR. 2. Parallelograms upon the same or equal bases are to one another as their altitudes.

THEOREM VII.

The measure of a triangle is half the product of its base multiplied by its altitude.

COR. 1. Triangles of the same or equal altitude are to one another as their bases.

COR. 2. Triangles on the same or equal bases are to one another as their altitudes.

DEF. A four-sided figure two of whose sides are parallel is called a trapezium.

THEOREM VIII.

The measure of a trapezium is the product of the altitude and half the sum of the parallel sides.

Let ABCD be a trapezium, of which AB, CD, are the parallel sides.

Draw the diagonal BD, and through A draw AE parallel to BD, cutting CD produced in E. Join BE.

because they are on the same base and between the same

parallels.

Add to each the triangle BCD.

=

Then the trapezium ABCD the triangle BEC.

But the measure of the triangle BEC is EC × BP (BP being the altitude).

And EC-ED+DC=AB+ CD since ABDE is a parallelogram.

Therefore the measure of the trapezium

This proposition may be applied to obtain the area of a

convex polygon. The figure shews the necessary construction, by which the polygon is divided into triangles and rectangular trapeziums.

THEOREM IX.

The square on the sum AC of two straight lines AB, BC is equal to the squares on AB and on BC, together with twice the rectangle AB,

BC.

a be the measure of AB,

B C

A

Let