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This may be done by means of the protractor. For example, let it be required to draw an angle equal to LABC (Fig. 122). Measure ZABC

C and write the number of degrees it contains inside of the angle, near the vertex.

B4

-A

Fig. 122 Draw a line as DEF (Fig. 123), and by the method of $65 draw

Fig. 123 on DF at the point E an angle containing the same number of degrees as ABC.

This is the required angle.

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EXERCISES

1. With a protractor and ruler draw a triangle having one right angle. What is the relation between the acute angles? Express the result in the form of an equation.

2. Using the protractor to draw the right angles, draw a square whose sides are 3 cm. long.

3. Draw a rectangle (Fig. 9) having two consecutive sides equal to 2 and 4" respectively.

4. Draw two intersecting lines, as AB and CD (Fig. 124). Mark a point E on CD. At E draw

C an angle x' equal to x.

5. Make the designs shown in Fig. 121.

A

B 6. Draw a triangle having two

D equal angles. To test the accuracy

Fig. 124 of your drawing measure the two sides opposite the equal angles. Compare them as to length.

This exercise shows that if two angles of a triangle are equal the sides opposite them are equal.

7. In the making of trusses for bridges, the beams are put together in the form of triangles having two equal sides. Make the design (Fig. 125). Lay off AB=BC=CD. Then draw angles equal to 60 degrees at A, B, C, and D. If your drawing is well made, the triangles AEB, BFC, and CGD

have three equal sides, and the points E, F, G lie on the same straight line.

8. In a triangle two angles are equal. The side opposite one of them is 6

inches long. How long is Fig. 125

the side opposite the other?

9. Draw a triangle having each of two angles equal to 60 degrees. What must be the size of the third angle? Measure the third angle and thus test the accuracy of your drawing. Measure the sides of the triangle.

This exercise shows that if the three angles of a triangle are equal, the three sides of the triangle are equal.

10. Draw a triangle having one angle equal to 60 degrees and another equal to 30 degrees. What is the size of the third angle?

Measure to two decimal places the side opposite the 90-degree angle, and the side opposite the 30-degree angle. Find the ratio of the two sides.

The exercise shows the following: If in a right triangle the acute angles of which are 30 degrees and 60 degrees, the side opposite the с

90-degree angle is twice as long as the side opposite the 30-degree angle.

11. In the right triangle ABC (Fig.

126) the side opposite the 30-degree angle 300 А

is assumed to be 6 inches long. What Fig. 126

is the length of CB? 67. Isosceles triangle. Equilateral triangle. Right triangle. A triangle having two equal sides is an isosceles triangle. A triangle having three equal sides is equilateral. A triangle having a right angle is a right triangle. The side opposite the right angle is called the hypotenuse.

6")

PARALLEL LINES

a line

E

F

H

68. Meaning of parallel lines. Draw segment AB (Fig. 127) about 12 cm. long. Place the sharp points of your compass on the ruler, or

-K squared paper, A

-B 8 cm. a part.

C

Fig. 127 On AB lay off segment CD equal to 8 cm. in length.

At C and D draw lines CE and DF perpendicular to AB, using the protractor.

On CE and DF lay off two equal lengths, as CG and

Р

D

DH.

Draw line IK passing through G and H.
Line IK is said to be parallel to AB.
Measure angles DHK and CGI and show that
CG and DH are perpendicular to IK.

Select any point P on AB. Draw a line perpendicular to AB at P and denote the point where it intersects IK by Q. Measure PQ.

How does the length of PQ compare with that of CG; of DH?

Because the perpendicular PQ which was drawn at a point P selected anywhere on AB, or its extension, is equal to the fixed lengths CG or DH, the lines AB and IK are said to be everywhere equally far apart. Hence, they cannot meet however far they are extended.

If two lines are drawn in the same plane surface, and if they do not intersect, however far extended, they are called parallel lines. The word parallel means running alongside of each other.

By the distance between two parallel lines is meant the length of the perpendicular between them.

One of the properties of parallel lines is that they are everywhere equally distant.

69. Symbol for parallelism. The statement AB is parallel to CD is written briefly in symbols: AB ||CD.

EXERCISES

1. Point out parallel lines in the classroom.

2. On a cube, or on a rectangular block, point out parallel lines. 3. Point out parallel lines on squared paper; on the ruler.

4. On a cube point out two lines which do not meet and are not parallel.

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5. In the classroom point out two lines which do not meet and are not parallel.

6. Are the rails

in the adjoining picture parallel? Give reason for your answer.

Do they look parallel to you?

7. Considerable knowledge of parallel lines was developed by the primitive races in connection with the art of weaving, basketry, and pottery. Many of the decorative designs are based on parallel lines, taking the shapes of rectangles, parallelograms, and squares (Fig. 9). Try to collect designs in weaving, clothing, pottery, household implements, etc., which illustrate the use of parallel lines.

70. Drawing parallel lines. On squared paper draw two parallel lines AB and

FE CD (Fig. 128) and a line EF intersecting FAS

PB AB and CD. Measure an

ZbA gles a and b. EGI

TD] How do they compare as to size?

E The equality of angles a and

Fig. 128 b (Fig. 128) suggests the following methods of drawing parallel lines.

1. The triangle method. Place one side, AB, of the triangle ABC (Fig. 129) along EF. Draw a line along the side BC.

Move the triangle by sliding side AB along EF until it

B Adl takes the position A, B, C1.

BF

Fig. 129 Draw a line along B, C.

Then BC is parallel to B, C, because the angles at B and B, are equal.

2. The T-square method. Place the head of the T-square (Fig. 130) along one edge AB of the drawing board. Draw a line along the straight edge. Then move the

B head of the square downward

Fig. 130

EA!

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