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6. The angles BEF, EFD are called interior angles. By measurement with the protractor, or by laying off, with the bevel, two adjacent angles equal to them, show that the sum of BEF and EFD is 180°.
We may also prove this thus:
But GEB+ BEF=2 rt. angles;
< BEF+ / EFD = 2 rt. angles.
7. There is no difficulty in verifying by actual measurement, or in proving the following equalities:
LAEF+ EFC-2 rt. angles.
8. To draw a straight line through a given point A parallel to a given straight line BC.
Through A draw DAE, cutting
BC, and make the angle DAF equal to the angle AEC.
AF is parallel to BC. then be produced, if
Of course we could have drawn GA parallel to BC by making the angle GAE equal to the alternate angle AEC. The line through A parallel to BC can also be drawn without measuring any angle, as follows:
With A as centre and radius, say, of 2 inches, describe a portion of a circle cutting BC in D. Measure off on this a distance AE of 1 inch, so that E is the
middle point of AD. With centre E and any radius of sufficient length to reach BC, describe a portion of a circle cutting BC in F; and let the diameter of this circle, through F, meet the circle again in G. Then AG is parallel to BC. For the sides AE, EG of the triangle AEG are equal to the sides DE, EF of the triangle DEF. Also the angles AEG, DEF are equal. Hence these triangles are equal in all respects, and the angle GAE is equal to the angle EDF. AG is therefore parallel to BC.
A number of exercises should be given pupils in drawing lines through given points parallel to lines in given positions, using both the preceding methods. At the end of each construction the accuracy of the work may be tested with the parallel rulers, or with ruler and set-square (Ch. IV., 5), or by examining whether lines drawn perpendicular to each pair of parallels are equal in length. (See 9 and 10, following.)
For the most part, in future, in drawing parallel lines parallel rulers are to be used, or ruler and set-square (Ch. IV., 5).
9. A straight line which is perpendicular to one of two parallel lines, is also perpendicular to the other. The truth of this should be tested by A drawing with the set-square a line. perpendicular to one of the parallels, and examining, with the set-square,
whether it is also perpendicular to the other.
Of course this is only a particular case of the truth, that parallel lines have each the same direction with respect to any third line that cuts them.
Or we may prove it as follows: If DFE is a right
angle, then since DFE+FEB=2 rt. angles, FEB must also be a right angle.
10. Two parallel lines are, of A course, throughout their lengths at the same distance from one another. For, with the set-square
or protractor, draw lines EF, GH, . . . perpendicular to AB and CD. Then, if the dividers be adjusted to the length EF, they will be found to be adjusted to the other lengths GH, .
We may prove that this is always the case, as follows: EF and GH are parallel to one another because they have the same direction with respect to the third line AB (or CD).
EGF = [GFH, being alternate angles;
Side FG is common to the two triangles;
We have everywhere illustrations of this. Thus we say that an ordinary board or ruler, whose sides are parallel, is of the same width throughout its length.
11. The method of drawing a line parallel to another by sliding the set-square along the ruler (Ch. IV., 5) receives its justification in the first paragraph of § 3 of this chapter. The line EACF corresponds to the edge of the ruler; the lines AB, CD to the edge of the set-square in its two positions; and the angles EAB, ECD to the angle of the set-square in its two positions, the angle of the set-square being of course always the same.
It may be added that, in drawing parallel lines, some prefer the ruler and set-square to parallel rulers. The cost of an instrument is saved. If the edges of ruler
and set-square are perfectly straight, the method gives absolutely correct results. Parallel rulers possibly work more rapidly and conveniently.
1. Draw a line through A parallel to a line BC, as follows: AB, AC. With B as centre, and radius equal to AC, describe a circle. With C as centre, and radius equal to AB, describe a circle. Let D be the point where the circles intersect on the same side of BC as A. Then AD is parallel to BC.
Test with parallel rulers.
Examine the equality of alternate angles.
Examine whether the sum of interior angles on the same side is
Prove that the alternate angles ADB, DBC are equal, and that, therefore, the lines are parallel.
2. If AB, CD intersect in O, and AO=OB, and CO=OD, what position do AD, CB occupy with respect to each other; and what do AC and DB? Apply tests with instruments. Give reasons, i.e., proof. 3. If AB, CD intersect in O, and AO=OD, CO OB, what position do AD, CB occupy with respect to each other?
4. Construct a quadrilateral with two sides equal, and the other two parallel and unequal.
5. In the receding question produce the equal sides to meet, and by applying tests determine the character of the two triangles so formed.
6. The two interior angles on the same side which one line makes with two others are 105° and 70°. Infer from 6 of Ch. VI. that the lines meet. On which side of the cutting line, and why?
7. AD and BF are parallel lines. From A draw equal lines AB, AC to BF; and also equal lines DE, DF, less than the former. Show that AC meets DE and DF on one side of the parallel lines, and AB meets them on the other side.
8. A, B are the extremities of the diameter of a circle, and parallel lines AC, BD are drawn, terminated by the circle. What is the relation of AC, BD as to magnitude? Give reasons.
9. A is a point not lying in the straight line BC. From A draw
as to the positions of the points K, L, M, ? Give reasons.
10. Two parallel lines are 3 inches apart, and a point A is taken 2 inches from one line and 1 inch from the other. Lines are drawn
through A terminated by the parallels.
how these lines are divided at A.
By measurement determine
11. Construct a triangle with sides 2, 3 and 4 inches. Bisect the sides and join the points of bisection. What do you observe as to the direction of the sides of the new triangle? What as to magnitudes of its angles and sides?
12. Construct a triangle, and through its angular points, with the parallel rulers draw lines parallel to the opposite sides. Four new triangles are thus constructed. Compare their sides and angles with those of the original triangle, and give results of comparison.
13. Construct a triangle ABC, and through any points D, E, F in the plane of the paper draw lines parallel to BC, CA, AB. Compare the angles of the new triangle with those of the original.
14. If through a point A any two lines be drawn, and through any point B lines be drawn parallel to the former two, prove that the angles at A and B are equal.
15. ABC, CDE are two triangles with AB, CD equal and parallel, and also BC, DE equal and parallel. What position do AC, CE occupy with respect to each other?
16. Make an irregular drawing on the paper to represent a pond, or other obstruction, and on opposite sides of it take points A and B. By a line construction about the pond, with measurements, obtain a line at A which if produced would pass through B, without placing the ruler on AB.
17. Draw two lines, both parallel to the same straight line. What is their position with respect to each other?
18. The side BC of a triangle ABC is produced to D. Bisect the angles BAC, ACD. Can the bisecting lines be parallel to one another?
19. On any line AC as diagonal, construct a quadrilateral ABCD with its opposite sides equal. How are the opposite sides placed with respect to each other? Test and give proof.
20. Two lines make an angle of 63° with each other. straight line 2 inches long with its ends resting on them, and making an angle of 80° with one of them.