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An obtuse angle is greater than a right angle.

An acute angle is less than a right angle.

THEOREM I.

If any straight line CD meet another straight line AB,

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the angles CDA, CDB are together equal to two right angles.

For they fill exactly the same space.

Conversely, if three straight lines AD, CD, BD meet in one point D so as to make the angles CDA, CDB equal to two right angles,

Then must AD and DB be in the same straight line.
Produce AD to E in the same straight line.
Then
CDA + CDE= 2R.

But

CDA + CDB= 2R (hypoth.).

Therefore

CDB = CDE.

Therefore DE and DB must have the same direction.

Therefore DB is in the same straight line with AB.

NOTE. When two angles are together equal to two right angles, either of these angles is called the supplement of the other.

THEOREM II.

If two straight lines AB, CD cut one another, the vertical

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angles must be equal, viz. AEC to DEB, and AED to BEC. For

AEC+ AED=2R,

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THEOREM III.

Any straight line AB is less than the two straight lines AC, CB which have the same extremities A and B.

For AB is the shortest line that can be drawn from A to B.

А

B

THEOREM IV.

Two straight lines AC, CB are less than two straight

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Much more then AC + CB < AD+DB.

Cor. Similarly, it may be shewn that any convex broken

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line ACDEB is less than another broken line AFGB by which it is enclosed.

THEOREM V.

The perpendicular is the shortest line that can be drawn

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from a point A to a straight line BC; and of the others AE which is nearer to the perpendicular is always less than one more remote AF.

Let the figure AFD be turned about BC so that A may fall on A'. Then, since ADB is a right angle,

ADB + BDA'=2R.
Therefore ADA' is a straight line.
Whence
AA <AE + EA:

:. 1. 3.

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Straight lines AE, AF; equally remote from the perpendicular AD, are equal.

{AE, AF are equally remote from AD if DE = DF).

Let the figure ADE be turned about AD; then since

ADE= ADF,
DE will take the direction of DF.
And since
DE=DF,

A
the point E will coincide with
the point F.

Therefore AE will coincide with AF, and must be equal

B E D to it.

Cor. 1. The angle AED will coincide with the angle AFD and must be equal to it.

Cor. 2. The angle EAD will coincide with the angle FAD and must be equal to it.

Conversely, if AE, AF be equal, they must be equally remote from the perpendicular.

For (Prop. v), neither can be nearer to the perpendicular than the other.

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