Ex. 1. Draw an acute angle and also an obtuse angle, and then compare them as above.

Ex. 2. Draw a small acute angle and a large acute one, and then compare them as above.

Ex. 3. Draw a small acute angle, and then draw another angle 3 times as large.

Ex. 4. Draw an acute angle, and also a right angle, and compare them as above.

SUG.-Article (39) shows how to draw a right angle.

Ex. 5. Draw any angle, and then draw another equal to it.

Ex. 6. Show that the angles a, b, and c are respectively,, and .6 of a right angle.*

Ex. 7. Show that angles a and b, Fig. 39, are respectively 13 and 11⁄2 times a right angle.

Ex. 8. Draw a regular inscribed hexagon, as in Fig. 31, and then comparing any one of its angles with a right angle, find that it is 1 times a right angle.

* Of course, absolute accuracy is not to be expected in such solutions.

62. An Inscribed Angle is an angle whose vertex is in the circumference of a circle, and whose sides are chords, as A, Fig. 41.

63. Theorem.-An inscribed angle is measured by one-half the arc included between its sides.

ILL.-The meaning of this is that an inscribed angle like A, which includes any particular arc, as cd, is only half as large as an angle would be at the centre, as cod, whose sides included the same arc, cd, or an equal arc. Thus, in this case, drawing the arc ab from A as a centre, with the same radius, Od, as cd is drawn with, I find that ab which measures A is of cd which measures cod.

Ex. 1. Which of the angles a, b, c, d, e is the largest? measured by? What b? What c? What d? What e?

What is a

Fig. 42.

Ex. 2. Which is the greatest angle, a, b, or c, Fig. 43? By what is a measured? By what b? By what c? What is the measure of a right angle? [See Example 10 in the preceding set.]

Ex. 3. Suppose I take a square card like CEDF, with a hole in one corner as at C, and sticking two pins firmly in my paper, as at A and B, place the corner of the card between them, as in Fig. 44, and then, keeping the sides of the card snug against the pins, put a

pencil through the hole c and move it around to A and then back to B; what kind of a line will the pencil trace? Will it make any difference whether c is a right angle or not? If any difference, what?

Ex. 4. By what part of a circumference is an angle of a regular inscribed hexagon measured? See (55), and Fig. 31. right angles is the angle of the hexagon equal to? sum of the six angles equal to?

How many What is the Ans. to last, 8 right angles.

Ex. 5. Show, from the way in which an equilateral triangle is constructed in Fig. 31, that one of its angles is measured by of a circumference, and hence is of a right angle.

64. Theorem.- When two lines intersect, they form either four right angles, or two equal acute and two equal obtuse angles.

ILL.-[The pupil can illustrate this for himself by drawing lines and noticing what angles are equal.]

EF

N

Ex. 1. Having a carpenter's square, an instrument represented by MON, I wish to test the angle O and ascertain whether it is, as it should be, a right angle. I draw an indefinite right line AB, and placing the angle O at some point C on this line with ON extending to the right on CB, I draw a line along OM. Turning the square over so that ON shall lie on CA, I draw another line along OM. Three cases may occur.-1st. Suppose the first line B drawn along OM is CF, and the second CE; what kind of an angle is O? 2d. Suppose the first line drawn is CE and the second CF; what kind of an angle is o? 3d. Suppose the first and second lines drawn along OM coincide and are CD; what kind of an angle is O?

FIG. 45.

Ex. 2. Show that the sum of all the angles formed by drawing lines on one side of a given line, and to the same point in the line, is two right angles.

65. Prob.-To bisect a given angle.

SOLUTION.-I wish to divide the angle AOB into two equal parts, ¿ e., to

B

bisect it. With O, the vertex, as a centre, and any convenient radius, as Oa, I
strike an arc, as ba, cutting the sides of the angle.
Then from a and b as centres, with the same radius
in each case, I strike two arcs intersecting as at P.
Drawing a line through P and O, it bisects the
angle; i. e., the angle POA = angle BOP. [Let
the pupil try this by cutting out the angle AOB,
and then folding the paper along the line P, or cut-
ting it through in the line OP, and then putting one
angle on the other, and thus see if they do not fit.]

Ex. 1. Draw an angle equal to of a right angle.
SUG.-First draw a right angle and then bisect it.

Ex. 2. Draw an angle equal to of a right angle.

SUG.-Draw a circle. Inscribe an equilateral triangle. rule, as in (55).] Then bisect any angle of this triangle. right angle, since the whole angle is 3. See Ex. 9 (61).

FIG. 46.

A

[Do it neatly, by This will be of a

Ex. 3. How does it appear that the angle EDF, Fig. 31, is of a right angle?

X 66. Parallel Straight Lines are such as, lying in the same plane, will not meet how far soever they are produced either way.

X

ILL. The sides of this page are parallel lines, as are also the top and bottom. The lines in Fig. 47 are parallel.

67. Prob. To draw a line through a given point and parallel to a given line.

FIG. 47.

SOLUTION. I wish to draw a line through the point O and parallel to the line AB. [The pupil should first draw some line, as AB, and mark some point, as O.] Ic take O as a centre, and with a radius* greater than the shortest distance to AB, as Oa, draw an indefinite arc aP. Then with a as a centre, and the same radius, I draw an arc from O to the line AB at b. Taking the distance Ob (the chord) in the dividers, I put the sharp point on a and strike a small arc intersecting this indefinite arc, as at P. Fi nally, drawing a line through O and P, it is the parallel sought.

A

a

FIG. 48.

B

This means "put the sharp point of the dividers on ○ and open them till the distance be tween the points (the radius) is more than the distance from O to AB."

68. Theorem.-Two parallel lines are everywhere the same distance apart.

ILL.-Let AB and CD be two parallel lines. I will examine them at the two

A

P

B

D

M

N

FIG. 49.

points O and P. To find how far apart the lines are at these points I draw the perpendiculars OM and PN. [The pupil should not guess at these, but actually draw them as instructed in (44).] Measuring these, I find them equal.

We can understand that this proposition must be true, since the lines could not approach each other for awhile and then separate more and more without being crooked; or, if they kept on approaching each other, they would meet after awhile, and so not be parallel.

A

69. Theorem.-Parallel lines make no angle with each other. ILL.-Let AB be a straight line, and suppose CD another straight line

FIG. 50.

passing through the point O. Now let CD turn around, first into the position D'C', then into D"C", etc., all the time passing through O. It is evident that the angle which this line makes with the line AB is all the time growing less, i. e., a'<a, and a"<a'. It is also evident that this angle will become 0 when the lines become parallel; for it

becomes less and less all the time, but is always something so long as the lines are not parallel.

70. Theorem.-Parallel lines have the same direction with each other.

ILL. Thus, in Fig. 47, the parallel lines all extend to the right and left, i. e., in the same direction.

Ex. 1. How shall the farmer tell whether the opposite sides of his farm are parallel?

Ex. 2. If we wish to cross over from one parallel road to another, is it of any use to travel farther in the hope that the distance across will be less?

Ex. 3. If a straight line intersects two parallel lines, how many angles are formed? How many angles of the same size? May they all be of the same size? When? When will they not be all of the same size?