Essentials of Geometry

2. The sum of the angles of a quadrilateral is how many times the sum of the angles of a triangle? The sum of the angles of a pentagon is how many times as great? Of a hexagon?

EXERCISES.

1. The sum of the interior angles of a quadrilateral equals what? 2. In the equiangular hexagon, each angle equals what?

3. If one of the angles of a parallelogram is 120°, how many degrees are there in each of the other angles?

4. If either of two adjacent angles on the same side, formed by one straight line meeting another, is a right angle, the other also must be a right angle. (Prove by the reductio ad absurdum.)

Suggestion. If the other angle is not a right angle, it is either greater or less. If greater, etc.

5. If two straight lines are parallel to a third line, they are parallel to each other.

Suggestion.-Let A and B be separately parallel to C; then will, etc. For draw DE perpen- Cdicular to C.

6. If two triangles have two sides of the one equal to two sides of the other, each to each, but the third sides unequal, the triangle which has the

greater third side will have the greater included angle.

Suggestion.-Construct a diagram that will conform to the Hypothesis,

and use the reductio ad absurdum. (See Th. VIII.)
7. If two straight lines cut a third line, mak-
ing the sum of the interior angles on the same
side less than two right angles, the two lines will
meet if sufficiently produced.

Suggestion.-Suppose they do not meet; then, by definition, they must be parallel; but if parallel, etc. (See Th. X, Cor. I.)

8. The diagonals of a parallelogram bisect each other. (See Th. X, Ex. 1.)

9. A parallelogram is a rectangle if its diagonals are equal.

10. The diagonals of a rhombus or square bisect each other at right angles.

11. A quadrilateral is a rhombus or square, if its diagonals bisect each other at right angles.

12. In a right triangle, the middle point of the hypotenuse is equally distant from the three vertices.

13. Every point in the bisector of an angle is equally distant from the sides of the angle.

CHAPTER III.

RATIO AND PROPORTION.

DEFINITIONS.

1. Ratio is the relation between two quantities of the same kind, and is expressed by the quotient of the first divided by the second.

2. The two quantities compared are called the Terms of the ratio; the first, the Antecedent, and the second, the Consequent. The antecedent and consequent together constitute a Couplet.

3. The sign of a ratio is the colon. A ratio is indicated by placing the sign between its terms, or by writing the antecedent over the consequent.

4. A Proportion is an equality of ratios. The equality is indicated by the double colon.

Thus, a b c d is a proportion, and is read, a is to b as c is to d. It is equivalent to

The terms of a proportion are four. The first and second terms form the first couplet; the third and fourth, the second couplet.

5. The first and fourth terms are called the Extremes; the second and third, the Means.

The first and third terms are antecedents; the second and fourth, consequents.

6. Of four proportional quantities, the last is called a Fourth Proportional to the other three, taken in order.

Thus, if ab: cd, d is a fourth proportional to a, b, and c. 7. Three quantities are proportional when the ratio of the first to the second equals the ratio of the second to the third. Thus, ab: b: c.

The middle term, b, is here called a Mean Proportional between the other two; and c, a Third Proportional to a and b. 8. A Continued Proportion is one in which several ratios are successively equal to each other; as

a b c d :: e : f :: g: h, etc.

9. Four quantities are in proportion by Alternation when antecedent is compared with antecedent and consequent with consequent.

Thus, if a : b :: c: d, by alternation we have a : c :: b : d.

10. Four quantities are in proportion by Inversion when the consequents are taken for antecedents, and the antecedents for consequents.

Thus, if a b c d, by inversion we have b: a :: d: c.

11. Four quantities are in proportion by Composition when the sum of the terms of each couplet is compared with either antecedent or consequent.

Thus, if ab:: cd, by composition we have

a+bac + dc, or

a + b : b :: c + d d.

12. Four quantities are in proportion by Division when the difference of the terms of each couplet is compared with either antecedent or consequent.

Thus, if a bed, by division we have

13. Four quantities are in proportion by Composition and Division when the sum of the terms of each couplet is compared with their difference.

Thus, if a : b :: c: d, by composition and division we have

14. Equimultiples of several quantities are the products of those quantities by the same multiplier.

Thus na and n b are equimultiples of a and b.

Note.-The object of this chapter is to communicate the principles of proportion, if they are not known; to refresh the memory, if they are; and to furnish a convenient reference when needed. The student should fix clearly in his mind the idea that a proportion is an equation -- that the expression a : b :: c: d means simply = and conversely.

α с

If four quantities are in proportion, the product of the extremes equals the product of the means.

Let a b c d; then will a × d

For

= bxc.

(1) by definition; and clearing of fractions, (2) ad=bc.

Therefore, etc.

THEOREM II.

If the product of two quantities equals the product of two other quantities, two of them may be made the means, and the other two the extremes of a proportion.