The angles at the base of an isosceles triangle are equal to one another, and, if the equal sides be produced, the angles on the other side of the base shall be equal to one another.

Let ABC be an isosceles triangle, having the side AB equal to the side AC, and let the straight lines AB, AC be produced to D and E, the angle ABC shall be equal to the angle ACB, and the angle CBD to the angle BCE.

C

In BD take any point F, and from AE, the greater, cut off.

AG equal to AF, the less,

and join FC, GB.

Because AF is equal to AG (const.) and AB to AC,

[I. 3.

[HYP.

the two sides FA, AC are equal to the two sides GA, AB, each to each,

and they contain the angle FAG common to the two triangles AFC, AGB,

therefore the base FC is equal to the base GB, the angle ACF is equal to the angle ABG, and the angle AFC is equal to the angle AGB.

[I. 4.

Again, because the whole AF is equal to the whole AG,

[CONST.

[HYP.

and the part AB is equal to the part AC,
therefore the remainder BF is equal to the remainder CG.

Now, FC has been shown to be equal to GB,

[AX. 3.

and the angle BFC (contained by BF, FC) to the angle
CGB (contained by CG, GB);

therefore the angle CBF is equal to the angle
BCG,

and the angle BCF is equal to the angle CBG.

·[I. 4.

But the whole angle ACF has been shown to be equal to the

whole angle ABG,

as well as the part BCF to the part CBG;

therefore the remaining angle ACB is equal to the remaining angle ABC.

[AX. 3.

COROLLARY.-Hence every equilateral triangle is also equiangular.

A corollary is a proposition, the truth of which follows immediately or may easily be deduced from what has been demonstrated in the proposition to which it is attached..

PROPOSITION 6. THEOREM.

If two angles of a triangle be equal, the sides also which subtend, or are opposite to the equal angles, shall be equal to one another.

Let ABC be a triangle, having the angle ABC equal to the angle ACB: the side AC shall be equal to the side AB.

A

For if AC be not equal to AB, one of them must be greater than the other.

Let AB be the greater, and from it cut off BD, equal to AC

and the angle DBC (contained by DB, BC) equal to the angle ACB (contained by AC, CB);

[HYP.

COROLLARY.-Hence every equiangular triangle is also equilateral.

NOTE.

A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom.

Two theorems are said to be converse, each of the other, when the hypothesis of each is the conclusion of the other (Syllabus).

Thus each of the two theorems 5 and 6 is the converse of the other. The student should notice that the converse of a true theorem may or may not be true.

Ex. 2.—In the figure of Prop. 5, if H is the point where BG cuts CF, BH is equal to HC.

Also FH is equal to HG.

PROPOSITION 7. THEOREM.

On the same base and on the same side of it there cannot be two triangles having their sides, which are terminated in one extremity of the base equal to one another, and likewise those which are terminated at the other extremity equal to one another.

On the same base AB, and on the same side of it, let there

be two triangles ACB, ADB, having their sides CA, DA, which are terminated at A, equal to one another, then they cannot have also the sides CB, DB, which are terminated at B, equal to one another.

Join CD.

In the case in which the vertex of each triangle is without the other triangle;

Because AC is equal to AD,

[HYP.

therefore the angle ACD is equal to the angle ADC; [I. 5.

But the angle ACD is greater than the angle BCD,

E

B

[AX. 9.

therefore the angle ADC is also

greater than the angle BCD;

much more then is the angle BDC greater than the angle BCD,

therefore BC is not equal to

BD;

[I. 5.

But if one of the vertices, as D, be within the other triangle ACB,

produce AC, AD to E and F respectively.

Then because AC is equal to AD,

[HYP.

therefore the angle ECD is equal to the angle FDC;

[I. 5.