79. By the method of superposition we are able to compare magnitudes of the same kind. Suppose we have two angles,

BC and BA in the position shown by the dotted line BG, the angle DEF is less than the angle ABC; but if the side EF falls in the position shown by the dotted line BH, the angle DEF is greater than the angle ABC.

80. If we have the angles ABC and DEF (Fig. 20), and place the vertex E on B and the side ED on BC, so that the angle DEF takes the position CBH, the angles DEF and ABC will together be equal to the angle ABH.

If the vertex E is placed on B, and the side ED on BA, so that the angle DEF takes the position ABF, the angle FBC will be the difference between the angles ABC and DEF.

If an angle is increased by its own magnitude two or more times in succession, the angle is multiplied by a number.

Thus, if the angles ABM, MBC, CBP, PBD (Fig. 21) are all equal, the angle ABD is 4 times the angle ABM. Therefore,

Angles may be added and subtracted; they may also be multiplied by a number.

PERPENDICULAR AND OBLIQUE LINES.

PROPOSITION I. THEOREM.

81. All straight angles are equal.

Let the angles ACB and DEF be any two straight angles.

To prove that

Proof. Place the

ACB on the DEF, so that the vertex

C shall fall on the vertex E, and the side CB on the side EF.

Ax. 7

82. COR. 1. All right angles are equal.

83. COR. 2. At a given point in a given line there can be

but one perpendicular to the line.

For, if there could be two Is, we should have

rt. of different magnitudes; but this is impossible, § 82.

84. COR. 3. The complements of the same angle or of equal angles are equal.

Ax. 3

85. COR. 4. The supplements of the same angle or of equal angles are equal. Ax. 3

NOTE. The beginner must not forget that in Plane Geometry all the points of a figure are in the same plane. Without this restriction in Cor. 2, an indefinite number of perpendiculars can be erected at a given point in a given line.

PROPOSITION II. THEOREM.

86. If two adjacent angles have their exterior sides in a straight line, these angles are supplementary.

Let the exterior sides OA and OB of the adjacent angles AOD and BOD be in the straight line AB.

87. DEF. Adjacent angles that are supplements of each. other are called supplementary-adjacent angles.

Since the angular magnitude about a point is neither increased nor diminished by the number of lines which radiate from the point, it follows that,

88. COR. 1. The sum of all the angles about a point in a plane is equal to a perigon, or two straight angles.

89. COR. 2. The sum of all the angles about a point in a plane, on the same side of a straight line passing through the point, is equal to a straight angle, or two right angles.

PROPOSITION III. THEOREM.

90. CONVERSELY: If two adjacent angles are supplementary, their exterior sides are in the same straight line.

Let the adjacent angles OCA and OCB be supplementary.

To prove that AC and CB are in the same straight line.

Proof. Suppose CF to be in the same line with AC.

Then

OCA and OCF are supplementary,

§ 86 (if two adjacent angles have their exterior sides in a straight line, these

But

angles are supplementary).

OCA and OCB are supplementary.

OCF and OCB have the same supplement.

.. LOCF = ≤ OCB.

.. CB and CF coincide.

.. AC and CB are in the same straight line.

Hyp.

$ 85

§ 60

Q. E. D.

Since Propositions II. and III. are true, their opposites are true. Hence,

§ 33

91. COR. 1. If the exterior sides of two adjacent angles are not in a straight line, these angles are not supplementary.

92. COR. 2. If two adjacent angles are not supplementary, their exterior sides are not in the same straight line.

PROPOSITION IV. THEOREM.

93. If one straight line intersects another straight line, the vertical angles are equal.

(if two adjacent angles have their exterior sides in a straight line, these

94. COR. If one of the four angles formed by the intersection of two straight lines is a right angle, the other three angles are right angles.

Ex. 1. Find the complement and the supplement of an angle of 49°. Ex. 2. Find the number of degrees in an angle if it is double its complement; if it is one fourth of its complement.

Ex. 3. Find the number of degrees in an angle if it is double its supplement; if it is one third of its supplement.