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172. Quadrilaterals that are not parallelograms.
1. Two of the sides of a trapezoid are parallel and two are not parallel. In the trapezoid represented above the side ac is parallel to the side
2. No two of the bounding lines of a trapezium are parallel.
3. In the trapezoid represented above no one of the angles is a right angle. Name the angles that are greater than right angles; the angles that are less than right angles. 4. Draw a trapezoid two of whose angles are right angles. 5. Can you draw a trapezoid having one and only one right angle?
6. Draw a trapezium one of whose angles is a right angle. 7. Can you draw a trapezium having more than one right angle?
8. Every quadrilateral (trapezium, trapezoid, or parallelogram) may be divided into two triangles. Remember that the sum of the angles of two triangles is equal to four right angles. Observe that the sum of the angles of the two triangles is equal to the sum of the angles of the quadrilateral. So the sum of the angles of a quadrilateral is equal to four right angles.
173. MISCELLANEOUS REVIEW.
1. If two of the angles of a trapezoid are right angles and the third is an angle of 60°, how many degrees in the fourth angle? Draw such a trapezoid.*
2. If the sum of three of the angles of a trapezium is 298°, how many degrees in the fourth angle? Draw such a trapezium.*
3. If one of the angles of a triangle is an angle of 80°, and the other two angles are equal, how many degrees in each of the other angles? Draw the figures.*
4. If one of the angles of a quadrilateral is a right angle, and the other three angles are equal, what kind of a quadrilateral is the figure?
5. One of the angles of a quadrilateral is a degrees; another is degrees; the third is c degrees. How many degrees in the fourth angle? ‡
6. The smallest angle of a triangle is r degrees; another angle is 2 r degrees, and the third is 3 r degrees:
Find the value of r; of 2 x; of 3 x.
7. 643,265,245,350. Without performing a division tell whether this number is exactly divisible by 9; by 5; by 10; by 25; by 50; by 121; by 18; by 6; by 15; by 30; by 90; by 163.†
8. A number is made up of the following prime factors: 2, 2, 3, 3, 5, 7, 11. Is the number exactly divisible by 18? by 25? by 35? by 77? by 21? by 30? by 45? by 8? (P. 333.)
* It is not expected that this drawing will be accurate in its angular measurements - simply an approximation to accuracy, to aid the pupil in recognizing the comparative size of angles.
A careful study of pages 71-75 inclusive will enable the pupil to make the statement called for, with little hesitation.
See problem 3, page 70.
174. A fraction may be expressed by two numbers, one of them being written above and the other below a short horizontal line; thus, %, 17, 379
3 11 240
175. The number above the line is the numerator of the fraction; the number below the line, the denominator of the fraction.
176. KINDS OF FRACTIONS.
1. A fraction whose numerator is less than its denominator is a proper fraction.
4, 3, 14, are proper fractions.
2. A fraction whose numerator is equal to or greater than its denominator is an improper fraction.
8 8, 47, are improper fractions.
NOTE.—The fraction .7 is a proper fraction. 2.7 may be regarded as an improper fraction or as a mixed number. If it is to be considered an improper fraction it should be read, 27 tenths; if a mixed number, 2 and 7 tenths.
3. Such expressions as the following are compound fractions:
of, of, of 71.
4. A fraction whose numerator or denominator is itself a fraction or a mixed number, is a complex fraction.
are complex fractions.
5. Any fraction that is neither compound nor complex is a simple fraction.
6. A fraction whose denominator is 1 with one or more zeros annexed to it, is a decimal fraction.
3, 11, 34, are simple fractions.
NOTE 1.- The denominator of a decimal fraction may be expressed by figures or it may be indicated by the position of the right-hand figure of its numerator with reference to the decimal point. When the denominator is thus indicated, the fraction is called a decimal and is said to be written decimally.
NOTE 2. All fractions that are not decimal are called common fractions. A decimal fraction when not "written decimally" (or thought of as written decimally) is usually classed as a common fraction.
7. A complex decimal is a decimal and a common fraction combined in one number.
.71, .251, .0561, are complex decimals.
177. There are three aspects in which fractions should now be considered.
1. THE FRACTIONAL UNIT ASPECT.
The numerator tells the number of things and the denominator indicates their name. In the fraction there are 5 things (magnitudes) called sevenths. In the fraction & there are five fractional units each of which is one eighth of some other unit called the unit of the fraction.
NOTE. The function of the denominator is to show the number of parts into which the unit of the fraction is divided; the function of the numerator, to show the number of parts (fractional units) taken.
II. THE DIVISION ASPECT.
The numerator of a fraction is a dividend; the denominator, a divisor, and the fraction itself, a quotient: thus, in the fraction ğ, the dividend is 5; the divisor, 8, and the quotient, §.
NOTE. In the case of an improper fraction, as, it may be more readily seen by the pupil that the numerator is the dividend, the denominator the divisor, and the fraction (= 2) the quotient; but the division relation is in every fraction, whether proper or improper, common or decimal, simple or complex.
III. THE RATIO ASPECT.
The numerator of a fraction is an antecedent; the denominator, a consequent, and the fraction itself, a ratio: thus, in the fraction, 7 is the antecedent, 10 the consequent, and the ratio.
NOTE 1.- This relation may be more readily seen by the pupil in the case of an improper fraction. In the fraction 4, 12 is the antecedent; 4, the consequent; 2, or 3, the ratio.
NOTE 2. - Every integral number as well as every fraction is a ratio. The number 8 is the ratio of a magnitude that is 8 times some unit of measurement to a magnitude that is 1 time the same unit of measurement. 8 (units of measurement) is the antecedent; 1 (unit of measurement) is the consequent, and the pure number 8 is the ratio.
178. REDUCTION OF FRACTIONS.
1. The numerator and the denominator of a fraction are its terms.
2. A fraction is said to be in its lowest terms when its numerator and denominator are integral numbers that are prime to each other.