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8. An angle is the amount of divergence between two lines that intersect, or meet; the point of meeting is called the vertex of the angle. Thus, in Fig. 6, the two lines form an angle whose vertex is at B. AAngles are distinguished by naming the vertex and a point on each line. Thus, in Fig. 6, the angle formed by the lines

FIG. 6.

A B and C B is called the angle ABC, or the angle CBA; the letter at the vertex is always placed at the middle. When an angle stands alone so that it cannot be mistaken for any other angle, only the vertex letter need be used. Thus, the angle referred to might be designated simply as the angle B. 9. A right angle is one of the angles formed by the intersection of two lines which are perpendicular to each other. In Fig. 7, the line A B is perpendicular to the line CD; therefore, the Cangles A B Cand A B Dare right angles.

10. An acute angle is less than a right angle. The angle ABC, Fig. 8, is an acute angle.

B

FIG. 7.

A

FIG. 8.

B

FIG. 9.

11.

An obtuse angle is

greater than a right angle. The angle A BD, Fig. 9, is an obtuse -D angle.

QUADRILATERALS.

12. A plane figure is any part of a plane, or flat, surface, bounded by straight or curved lines.

13. A quadrilateral is a plane figure bounded by four straight lines.

14. A parallelogram is a quadrilateral whose opposite sides are parallel.

There are four kinds of parallelograms: the rectangle, the square, the rhomboid, and the rhombus.

Diagonal

FIG. 10.

15. A rectangle is a parallelogram whose angles are all right angles. (Fig. 10.)

FIG. 11.

FIG. 12.

FIG. 13.

FIG. 14.

16. A square is a rectangle whose sides are all of the same length. (Fig. 11.)

17. A rhomboid is a parallelogram whose opposite sides are equal, and whose angles are not right angles. (Fig. 12.)

18. A rhombus is a rhomboid having equal sides. (Fig. 13.)

19. A trapezoid is a quadrilateral having only two of its sides parallel. (Fig. 14.)

20. The altitude of a parallelogram or trapezoid is the perpendicular distance between the parallel sides. The length of the dotted lines in Figs. 12, 13, and 14 is the altitude. 21. The base of a quadrilateral is the side on which it is supposed to stand. Any side may be taken as the base.

22. The area of a plane figure is the number of square units contained in its surface. The square unit may be a square inch, square foot, square yard, square meter, etc., as is most convenient.

23. The area of a parallelogram is equal to the product of the base and the altitude. This can be shown readily in

the case of the rectangle. Suppose, for example, the leaf of a book is 6 inches wide and 9 inches long (Fig. 15). It is a

9

FIG. 15.

rectangle with a base of

9 inches and an altitude of 6 inches. Suppose the base to be divided into 9 equal parts, each 1 inch in length, and assume lines to be drawn through each point of division, parallel to the short sides of the rectangle. In a similar

manner, suppose the altitude, or short side, to be divided into 6 equal parts, each 1 inch long, and through these points of division let lines be drawn parallel to the base. The rectangle is divided by these two sets of lines into little squares, as shown in Fig. 15. The area of one of the small squares is 1 square inch, since each of its sides is 1 inch in length. There are 9 of the squares in each horizontal row, and there are 6 rows. Hence, the total number of the little squares is 6X9 = 54, and the area of the surface is 54 square inches.

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24. Rule. To find the area of a rectangle, multiply the base by the altitude.

25. In ordinary language, the base and altitude of a rectangular surface are spoken of as length and breadth; the area of the surface is obtained by multiplying together the length and breadth. In applying the above rule, care must be taken that the base and altitude, or length and breadth, are reduced to the same kind of units. For example, if the base is given in feet and the altitude in inches, they cannot be multiplied together unless both are feet or both inches. This principle is of great importance, and holds good throughout the subject of Mensuration.

It must not be understood from the foregoing that feet can be multiplied by feet or inches by inches. In multiplication the multiplier is always abstract. In Fig. 15 there are 9 square inches in 1 row, and 6 times as many in 6 rows.

The operation in reality is 9 sq. in. x 6 54 sq. in., or 6 sq. in. X9 54 sq. in.

26.

feet wide?

EXAMPLE.-What is the area of a floor 16 feet long and 131

SOLUTION. The base is 16 feet and the altitude is 13 feet.

Area = base X altitude = 16×13 216 sq. ft. Ans.

27. The area of any parallelogram is equivalent to the area of a rectangle of the same base and altitude. In Fig. 16, the plane figure A BDC is a rhomboid. Suppose the corner ACE is cut off, as shown, and placed at the other end in the position BDF. If the cutting line A E is perpendicular to

FIG. 16.

the base CD, the new figure ABFE is a rectangle. It is plain that the base and altitude of the rectangle are the same as the base and altitude of the rhomboid, and that the areas of the two figures are the same.

28. To find the area of any parallelogram, multiply the base by the altitude.

EXAMPLE. A piece of cloth 1 yard wide is "cut on the bias," that is, it has the shape shown in Fig. 16. If the length of the strip is 8 feet, what is its area?

SOLUTION. The altitude is 1 yd.
Area = base X altitude =

3 ft., and the base is 8 ft. Hence, 8×3 = 24 sq. ft. Ans.

29. Rule. To find the area of a trapezoid, multiply onehalf the sum of the parallel sides by the altitude.

30. The reason for this rule will appear from an examination of Fig. 17. If E and F be the middle points of the sides that are not parallel, and if AES and BF2 be cut off below by 4-3 and 1-2, perpenB dicular to AB, and placed above, as shown, we have a

FIG. 17.

rectangle whose area is equal to EFX HI. But EF is as much less than A B as it is greater than DC. In other words,

EF half the sum of the parallel sides of the trapezoid.

Hence, area of trapezoid =

DC+AB
2

XHI.

EXAMPLE.-A piece of land has the form of a trapezoid. The parallel sides are, respectively, 40 rd. and 56 rd. long, and the perpendic ular distance between them is 35 rods. How many acres are contained in the piece?

SOLUTION.-One-half the sum of the parallel sides

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=

40+56
2

= :48 rd.

= 10.5 acres. Ans.

31. The perimeter of a quadrilateral is the sum of the lengths of its four sides.

EXAMPLE.-A room is 23 ft. long and 18 ft. wide. What is its perimeter ?

SOLUTION.-Perimeter = 23+23+18+18 = 23 X 2 + 18 x 2 = 82 ft.

PLASTERING, PAINTING, AND KALSOMINING.

Ans,

32. Plastering, painting, and kalsomining are usually estimated by the square yard. Allowances for doors, windows, etc. are not regulated by any established usage. Sometimes no deduction is made for them, sometimes onehalf their extent is deducted; but this is a matter usually specified in the contract.

33. EXAMPLE.-At 22 cents per square yard, what will it cost to plaster a room 65 ft. long, 22 ft. wide, and 15 ft. high; deducting in full for 8 doors 4 ft. 6 in. wide and 11 ft. 6 in. high; 10 windows 3 ft. 6 in. wide and 8 ft. high; and a baseboard 6 in. high extending around the room?

SOLUTION.-Perimeter of the room =

Area of walls..

Area of ceiling..

Total

65 × 2+22 × 2 = 174 ft. = 174 X 15 = 2,610 sq. ft.

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41 X 11 X 8 =
= = 31 X 8 X 10
(perimeter less the width

414 sq. ft.

=

280 sq. ft.

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