by a point from either end of it, and therefore every straight line has two directions, which are the opposite of each other. We speak of the direction from A to B as the direction AB, and of the direction from B to A as the direction BA.

B

One line meeting another at some other point than the extremity, makes two angles with it. Thus the angle BDF is the difference in the directions DB and DF; and the angle BDC is the difference in the directions DB and DC.

C

D

F

When two lines pass through or cut each other, four angles are formed, each direction of one line making a difference with each direction of the other.

The opposite angles formed by two lines cutting each other are called VERTICAL angles.

A line which cuts another, or which cuts a figure, is called a SECANT.

PROBLEMS ON ANGLES.

87. Angles may be compared by placing one upon the other, when, if they coincide, they are equal. Problem.-One angle may be added to another.

Let the angles ADB and BDC be adjacent and in the same plane. The angle ADC is plainly equal to the sum of the other two (9).

D

A

Problem. An angle may be subtracted from a greater

one.

For the angle ADB is the difference between ADC and BDC.

It is equally evident that an angle may be a multiple or a part of another angle; in a word, that angles are quantities which may be compared, added, subtracted, multiplied, or divided.

But angles are not magnitudes, for they have no extent, either linear, superficial, or solid.

ANGLES FORMED AT ONE POINT.

88. Theorem.-The sum of all the successive angles formed in a plane upon one side of a straight line, is an invariable quantity; that is, all such sums are equal to each other.

If AB and CD be two straight lines, then the sum of all the successive angles at E is equal

to the sum of all those at F.

For the line AE may be placed on CF, the point E on the point F. Then EB will fall on FD, for when two straight lines coincide in part, they must coincide throughout their mutual extent (51). Therefore, the sum of all

E

B

C

F

D

the angles upon AB exactly coincides with the sum of all the angles upon CD, and the two sums are equal. 89. When one line meets another,

making the adjacent angles equal, the angles are called RIGHT ANGLES.

One line is PERPENDICULAR to the other when the angle which they make is a right angle.

Two lines are OBLIQUE to each other

when they make an angle which is greater or less than a right angle.

90. Corollary.—All right angles are equal.

For each is half of the sum of the angles upon one side of a straight line. By the above theorem, these sums are always equal, and (7) the halves of equal quantities are equal.

91. Corollary. The sum of all the successive angles formed in a plane and upon one side of a straight line, is equal to two right angles.

92. Corollary.-The sum of all the successive angles formed in a plane about a point, is equal to four right angles.

93. Corollary.-When two lines cut each other, if one of the angles

thus formed is a right angle, the other three must be right angles.

94. In estimating or measuring angles in geometry, the right angle is taken as the standard.

An angle less than a right angle is called ACUTE.

An angle greater than one right angle and less than the sum of two, is called OBTUSE. Angles greater than the sum of two right angles are rarely used in elementary geometry.

When the sum of two angles is equal to a right angle, each is the COMPLEMENT of the other.

When the sum of two angles is equal to two right angles, each is the SUPPLEMENT of the other.

95. Corollary-Angles which are the complement of the same or of equal angles are equal (7).

96. Corollary.-Angles which are the supplements of the same or of equal angles are equal.

97. Corollary.-The supplement of an obtuse angle is acute.

98. Corollary. The greater an angle, the less is its supplement.

99. Corollary.-Vertical angles

are equal. Thus, a and i are each supplements of e.

e

a

100. Theorem.-When the sum of several angles in a plane having their vertices at one point is equal to two right angles, the extreme sides form one straight line.

If the sum of AGB, BGC, etc., be equal to two right angles, then will AGF be one straight line.

A

B.

C

D

F

For the sum of all these angles being equal (91) to the sum of the angles upon one side of a straight line, it follows that the two sums may coincide (40), or that AGF may coincide with a straight line. Therefore, AGF is a straight line.

EXERCISES.

101. Which is the greater angle,

a or b, and why?

What is the greatest number of points

in which two straight lines may cut

each other? In which three may cut each other? Four? 102. The student should ask and answer the question "why" at each step of every demonstration; also, for every corollary. Thus:

Why are vertical angles equal? Why are supplements of the same angles equal?

And in the last theorem: Why is AGF a straight line? Why may AGF coincide with a straight line? Why may the two sums named coincide? Why are the two sums of angles equal?

PERPENDICULAR AND OBLIQUE LINES.

103. Theorem.-There can be only one line through a given point perpendicular to a given straight line.

For, since all right angles are equal (90), all lines lying in one plane and perpendicular to a given line, must have the same direction. Now, through a given point in one direction there can be only one straight line (49).

Therefore, since the perpendiculars have the same direction, there can be through a given point only one perpendicular to a given straight line.

When the point is in the given line, this theorem must be limited to one plane.

104. Theorem.-If a perpendicular and oblique lines fall from the same point upon a given straight line, the perpendicular is shorter than any oblique line.

If AD is perpendicular and AC oblique to BE, then AD is shorter than AC.

Let the figure revolve upon BE as upon an axis (53), the point A falling upon F, and the lines AD and AC upon FD and FC.

B D

A

E

Now, the angle CDF is equal to the angle CDA, and both are right angles. Therefore, the sum of those two angles being equal to two right angles (100), ADF is a straight line, and is shorter than ACF (54). Therefore, AD, the half of ADF, is shorter than AC, the half of ACF.

105. Corollary.-The distance from a point to a straight line is the perpendicular let fall from the point to the line.