as many times as the polygon has sides.

But the sum of

the interior angles is equal to two right angles taken as many times as the polygon has sides, less two: hence, the sum of the exterior angles is equal to two right angles taken twice; that is, equal to four right angles; which was to be proved.

PROPOSITION XXVIII. THEOREM.

In any parallelogram, the opposite sides are equal, each to \ each.

Let ABCD be a parallelogram: then will AB be equal to DC, and AD to BC.

For, draw the diagonal BD. Then, because AB and DC are parallel, the angle DBA is equal to its alternate

angle BDC (P. XX., C. 2): and, because AD and BC are parallel, the angle BDA is equal to its alternate angle DBC. The triangles ABD and CDB, have, therefore, the angle DBA equal to CDB, the angle BDA equal to DBC, and the included side DB common; consequently, they are equal in all of their parts: hence, AB is equal to DC, and AD to BC; which was to be proved.

Cor. 1. A diagonal of a parallelogram divides it into two triangles equal in all their parts.

Cor. 2. Two parallels included between two other par allels, are equal.

Cor. 3. If two parallelograms have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, they will be equal.

If the opposite sides of a quadrilateral are equal, each to each, the figure is a parallelogram.

In the quadrilateral ABCD, let AB be equal to DC, and AD to BC: then will it be a parallelogram.

Then, the

A

Draw the diagonal DB. triangles ADB and CBD, will have

the sides of the one equal to the sides of the other, each to each; and therefore, the triangles will be equal in all of their parts: hence, the angle ABD is equal to the angle CDB (P. X., S.); and consequently, AB is parallel to DC (P. XIX., C. 1). The angle DBC is also equal to the angle BDA, and consequently, BC is parallel to AD: hence, the opposite sides are parallel, two and two; that is, the figure is a parallelogram (D. 28); which was to be proved.

PROPOSITION XXX. THEOREM.

If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

In the quadrilateral ABCD, let AB

D

be equal and parallel to DC: then will

the figure be a parallelogram.

A

Draw the diagonal DB. Then, because AB and DC are parallel, the angle ABD is equal to its alternate angle CDB. Now, the triangles ABD and CDB, have the side DC equal to AB, by hypothesis, the side DB common, and the included angle ABD equal to BDC, from what has just

been shown; hence, the triangles are equal in all their parts (P. V.); and consequently, the alternate angles ADB and DBC are equal. The sides BC and AD are, therefore, parallel, and the figure is a parallelogram; which was to be proved.

Cor. If two points be taken at equal distances from a given straight line, and on the same side of it, the straight line joining them will be parallel to the given line.

PROPOSITION XXXI. THEOREM.

The diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other.

A

C

Let ABCD be a parallelogram, and B AC, BD, its diagonals: then will AE be equal to EC, and BE to ED. For, the triangles BEC and AED, have the angles EBC and ADE equal (P. XX., C. 2), the angles ECB and DAE equal, and the included sides BC and AD equal: hence, the triangles are equal in all of their parts (P. VI.); consequently, AE is equal to EC, and BE to ED; which was to be proved

Scholium. In a rhombus, the sides AB, BC, being equal, the triangles AEB, EBC, have the sides of the one equal to the corresponding sides of the other; they are, therefore, equal: hence, the angles AEB, BEC, are equal, and therefore, the two diagonals bisect each other at right angles..

1. THE RATIO of one quantity to another of the same kind, is the quotient obtained by dividing the second by the first. The first quantity is called the ANTECEDENT, and the second, the CONSEQUENT.

2. A PROPORTION is an expression of equality between two equal ratios. Thus,

expresses the fact that the ratio of A to B is equal to the ratio of C to D. In Geometry, the proportion is written thus,

and read, A is to B, as C is to D.

3. A CONTINUED PROPORTION is one in which several ratios are successively equal to each other; as,

A B

C D E : F:: G : H, &c

4. There are four terms in every proportion. The first and second form the first couplet, and the third and fourth,

the second couplet. The first and fourth terms are called extremes; the second and third, means, and the fourth term, a fourth proportional to the other three. When the second term is equal to the third, it is said to be a mean proportional between the extremes. In this case, there are but three different quantities in the proportion, and the last is said to be a third proportional to the other two. Thus, if we have,

A: B :: B : C,

B is a mean proportional between A and C, and C is a third proportional to A and B.

5. Quantities are in proportion by alternation, when antecedent is compared with antecedent, and consequent with consequent.

6. Quantities are in proportion by inversion, when antecedents are made consequents, and consequents, antecedents.

7. Quantities are in proportion by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent.

8. Quantities are in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent.

9. Two varying quantities are reciprocally or inversely proportional, when one is increased as many times as the other is diminished. In this case, their product is a fixed quantity, as xy = m.

10

Equimultiples of two or more quantities, are the products obtained by multiplying both by the same quantity. Thus, mA and mB, are equimultiples of A and B.