hence, the triangles are equal in all respects (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 are taken at equal distances from a given straight line, and on the same side of it, the straight line joining them is parallel to the given line.

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

Let ABCD be a parallelogram, and AC, BD, its diagonals: then AE is equal to EC, and BE to ED.

For, the triangles BEC and AED, have the angles EBC and ADE equal (P. XX.,

B

C. 2), the angles ECB and DAE equal, and the included. sides BC and AD equal: hence, the triangles are equal in all respects (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 in all respects: hence, the angles AEB, BEC, are equal, and therefore, the two diagonals bisect each other at right angles.

EXERCISES.

1. Show that the lines which bisect (halve) two vertical angles, form one and the same straight line.

3. One of the two interior angles on the same side, formed by a straight line meeting two parallels, is one-half of a right angle; what is the other angle equal to?

4. The sum of two angles of a triangle is of a right angle; what is the other angle equal to?

5. One of the acute angles of a right-angled triangle

is of a right angle; what is the other?

6. Show Show that the line which bisects the exterior vertical angle of an isosceles triangle is parallel to the base of the triangle. (P. XXV., C. 6; P. XIX., C. 1.)

7. The sum of the in

B

E

-D

terior angles of a polygon is 12 right angles; what is the polygon?

8. What is the sum of the interior angles of a heptagon equal to?

9. The sum of five angles of a given equiangular polygon is 8 right angles; what is the polygon?

10. What part of a right angle is an angle of an equiangular decagon?

11. How many sides has a polygon in which the sum of the interior angles is equal to the sum of the exterior angles?

- 12. Construct a square, having given one of its diagonals.

NOTE 1.-The complement of an angle is the difference between that angle and a right angle; thus, EOB is the complement of AOE.

NOTE 2.-The supplement of an angle is the difference between that

E

angle and two right angles; thus, EOC is the supplement of AOE.

13. An angle is of a right angle; what is its complement? and what its supplement?

14. Show that any two adjacent angles of a parallelogram are supplements of each other.

15. Show that if two parallelograms have one angle in each equal, their remaining angles are equal each to each.

16. Show that if two sides of a quadrilateral are parallel and two opposite angles equal, the figure is a parallelogram.

17. Show that if the opposite angles of a quadrilateral are equal, each to each, the figure is a parallelogram.

- 18. Show that the lines which bisect the angles of any quadrilateral form, by their intersection, another quadrilateral, the opposite angles of which are supplements of each other. [Twice the angle B is equal to the sum of the angles CDE and DEF.]

+

B

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

the second, the CONSEQUENT.

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

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 three others. 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 two others. Thus, if we have,

B is a

A : B :: B: C,

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 with either antecedent or consequent.

9. Four quantities are reciprocally proportional, when the first is to the second as the fourth is to the third. Two varying quantities are reciprocally proportional, when their product is a fixed quantity, as xy = m.

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