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 ▲, having

A

ACB;

then shall the ▲ ABC be

L

also let the equal sides AB, AC be produced to G, H;

then shall -GBC be = L-HCB.

B

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;

G/

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.

H

Thus AB, BC would fall on the former positions of AC, CB;

.*. LABC is: LACB.

Also BG, BC would fall along the former positions of CH, CB;

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.

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

Then shall the ▲ş ABÇ, DEF be equal in all respects.

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

then will BA fall along ED, . LABC is = L DEF, and CA will fall along FD, . LACB is = LDFE; .. the point A will fall on the point D,

.. the triangle ABC will coincide with the ▲ 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.

For if the 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 ▲ ABC be respectively equal to the sides DE, EF, FD of the ▲ DEF.

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

Suppose the ▲ ABC taken up, reversed, and placed so that BC may fall on EF,

and A on the opposite side of EF to that on which D is, as H.

·· EH is=ED, :. LEHD is = LEDH;

(1. 2)

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

also BA, AC are respectively=ED, DF;

..AS ABC, DEF are equal in all respects. (Prop. 1.)