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PROPOSITION III.

If two angles and the adjacent side of one triangle are respectively equal to two angles and the adjacent side of another triangle, then shall the triangles be equal in all respects.

D

B

In the As ABC, DEF, let the ZS ABC, ACB and the adjacent side BC be respectively equal to the es DEF, DFE and the adjacent side EF.

Then shall the as ABC, DEF be equal in all respects.

For if - ABC be applied to a DEF so that BC may fall on EF;

then will BA fall along ED, ::: LABC is = 2 DEF, and CA will fall along FD, ::: ACB is = _DFE;

.: the point A will fall on the point D, .. the triangle ABC will coincide with the A DEF, and is .. equal to it in all respects.

PROPOSITION IV.

If two angles of a triangle are equal to one another, the sides opposite to those angles are equal.

B

Let ABC be a triangle, having

the ABC= the L ACB,
then shall the side AC be = the side AB.

For if the a ABC be taken up, reversed, and applied to the former position of BC and on the same side of it as before, so that C may fall on the former position of B and B on that of C; then will CA fall along that of BA, :;: ABC is = LACB, and BA along that of CA for the same reason ;

.. A will fall on its former position,
and .. AC will coincide exactly with that of AB,

.: AC.is= AB.

PROPOSITION V.

If the three sides of one triangle are respectively equal to the three sides of another, the triangles shall be equal in all respects.

Let the sides AB, BC, CA of the A ABC be respectively equal to the sides DE, EF, FD of the A DEF.

Then shall the As ABC, DEF be equal in all respects.

Suppose the A ABC taken up, reversed, and placed so that BC may fall om EF, and A on the opposite side of EF to that on which D is,

as H.

[blocks in formation]

:: EH is = ED, LEHD is = EDH; and .:: FH is = FD, L FHD is = LFDH; .: the whole EHF is = the whole 2 EDF,

and ...BAC is = . EDF.

(1. 2) (I. 2)

(ii) If HD passes through one extremity of EF as F.

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::: EH is = ED, .: LEHD is = . EDH;

and ... BAC is = LEDF.

(1. 2)

[blocks in formation]

::: EH is= ED, :LEHD is = 1 EDH; (1. 2) and :: FH is = FD, ... FHD is = FDH; (1. 2)

... EHF is = . EDF, and

...BAC is = 1 EDF. Thus in any case the

<BAC is = EDF, also BA, AC are respectively = ED, DF; .:AS ABC, DEF are equal in all respects. (Prop. 1.)

Note. Propositions I, III, and V hold if the triangles are situated thus :

[blocks in formation]

having the sides AB, BC, CA corresponding to DE, EF, FD.

In propositions I and III it will be necessary, after taking up ^ ABC, to reverse it before applying it to A DEF.

In proposition V, the A ABC must be applied without being reversed.

The demonstrations will be the same as before.

DEFINITIONS.

An equilateral triangle is one which has all its sides equal.

A circle is a plane figure contained by one line called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal. That point is called the centre of the circle, and those straight lines radii.

It is taken for granted that a circle can be described having any centre and radius equal to any given straight line.

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