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PLANE GEOMETRY.

BOOK I.

INTRODUCTION.

A point has no magnitude. It has position.
A line has length but not breadth.
The extremities of lines are points.

A straight line is one which lies evenly between its extremities.

It is taken for granted :

That a straight line may be drawn from any one point to any other point;

And also that a terminated straight line may be produced to any length in a straight line.

A superficies has length and breadth but not thickness. The extremities of superficies are lines.

A plane is a superficies in which any two points being taken the straight line joining them lies wholly in that superficies.

C. G.

An angle is formed by two lines drawn from a point.

The two lines are called the arms, and the point the vertex of the angle.

If the arms are in one plane the angle is called a plane angle, and if they are straight lines a plane rectilineal angle.

Whenever the term angle is used a plane rectilineal angle is meant unless otherwise expressed.

The size of an angle does not depend on the length of its arms but on their direction, and is said to be greater or less according as the difference of direction of its arms is greater or less...

It is evident that
Two straight lines-cannot enclose a space.
Hence it follows that
Two straight lines cannot have a common segment..

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For if two straight lines ABC, ABH could have a common segment AB; then the straight line ABC might be turned about its extremity A, towards the side on which BH is, so as to cut BH; and thus two straight lines would enclose a space, which is impossible.

Hence it follows that

If two straight lines pass through the same point they will coincide entirely or cut one another.

For if not, if possible let them fall otherwise as AOB, POQ having a common point 0,

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Then AOB might be turned about one extremity A towards the side on which OQ is, so as to cut OQ and also OP.

Thus two straight lines would enclose a space, which is impossible.

It is evident that

Magnitudes which may be made to coincidi, i.e. exactly fill the same space, are equal to one another.

If two angles can be so placed that their vertices coincide and the arms of one fall along the arms of the other, those angles are said to be equal to one another,

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If the angle BAC were taken up, reversed, and then applied to its former position, so that A fell on its former position and AC along that of AB; then would AB fall along that of AC.

STRAIGHT LINES, ANGLES AND TRIANGLES.

DEFINITION. A triangle is a plane figure contained by three straight lines.

PROPOSITION I. If two sides and the included angle of one triangle be respectively equal to two sides and the included angle of another triangle, the triangles shall be equal in all respects.

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In the as ABC, DEF, let the two sides AB, AC be respectively equal to DE, DF,

BAC = - EDF; then shall the as ABC, DEF be equal in all respects.

and

For since the <BAC is = the LEDF, the a ABC may be applied to the a DEF so that A may fall on D, and the straight lines AB, AC along DE, DF;

then B will fall on E, ::: AB is = DE,
and C will fall on F, ::: AC is = DF;

.. BC will coincide with EF. For if it fell otherwise, as EKF, then two straight lines would enclose a space, which is impossible;

.. BC is = EF. Also the A ABC will coincide with and is .:.= A DEF,

LABC will coincide with and is .:= DEF; so

LACB is = DFE.
Hence the as ABC, DEF are equal in all respects.

DEFINITION. An isosceles triangle is one which has two of its sides equal.

PROPOSITION II. The angles at the base of an isosceles triangle are equal; and if the equal sides be produced, the angles on the other side of the base shall be equal.

Let ABC be an isosceles A, having the side AB = the side AC, then shall the < ABC be = the < ACB; also let the equal sides AB, AC be produced to G, H;

then shall < GBC be = 2.HCB.

For if the figure were taken up, reversed, and then applied to its former position so that A might fall on its / former position and ABG along that of ACH;

then would ACH fall along that of ABG; also B would fall on that of C, :: AB is = AC,

and C on that of B for the same reason, .: BC would coincide with its former position. Thus AB, BC would fall on the former positions of AC, CB;

i LABC is = LACB. Also BG, BC would fall along the former positions of CH, CB;

.. GBC is = 1 HCB.

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