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1. Things which are equal to the same thing are equal to each other.

2. If equals are added to equals the sums are equal.

3. If equals are taken from equals the remainders are equal.

4. If equals are added to unequals the sums are unequal, and the greater sum is obtained from the greater magnitude. 5. If equals are taken from unequals the remainders are unequal, and the greater remainder is obtained from the greater magnitude.

6. Things which are double the same thing, or equal things, are equal to each other.

7. Things which are halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. The whole is equal to all its parts taken together.

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Let ZBCA and ▲ FED be any two straight angles.

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Proof. Apply the ▲ BCA to the ▲ FED, so that the vertex C shall fall on the vertex F, and the side CB on the side EF.

Then CA will coincide with ED,

(because BCA and FED are straight lines and have two points common). Therefore the BCA is equal to the Z FED.

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

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86. COR. 2. The angular units, degree, minute, and second, have constant values.

87. COR. 3. The complements of equal angles are equal.

88. COR. 4. The supplements of equal angles are equal.

89. COR. 5. At a given point in a given straight line one perpendicular, and only one, can be erected.

PROPOSITION II. THEOREM.

90. If two adjacent angles have their exterior sides in a straight line, these angles are supplements of each other.

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

To prove AOD + BOD=2 rt. 4.

Proof.

AOB is a straight line.

Hyp.

.. the ZAOB is a st. ▲, and equal to 2 rt. .

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91. SCHOLIUM. Adjacent angles that are supplements of each other are called supplementary-adjacent angles.

92. COR. 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,

The sum of all the angles about a point in a plane is equal to two straight angles, or four right angles.

The sum of all the angles about a point 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.

93. CONVERSELY: If two adjacent angles are supplements of each other, their exterior sides lie in the same straight line.

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Let the adjacent ▲ OCA + OCB = 2 rt. §.

To prove AC and CB in the same straight line.

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

Then

ZOCA +2 OCF 2 rt. 4,

(being sup.-adj. 4).

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But

ZOCA +2 OCB = 2 rt. 4.

Hyp.

:: ZOCA +2 OCF=▲ OCA + ≤ OCB.

Ax. 1

Take

Then

away from each of these equals the common ▲ OCA.

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

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Q. E. D.

94. SCHOLIUM. Since Propositions II. and III. are true, their opposites are true; namely,

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If the exterior sides of two adjacent angles are not in a straight line, these angles are not supplements of each other. If two adjacent angles are not supplements of each other, their exterior sides are not in the same straight line.

PROPOSITION IV. THEOREM.

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

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:.Z OCA +2 OCB = ≤ OCA + Z ACP.

Ax. 1

Take away from each of these equals the common ≤ OCA.

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96. 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,

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