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THEOREM (Euclid I. 5). Definition.-An isosceles triangle has two sides equal.
Repeat.-The enunciation of Euc. I. 4, and
angle are equal to one another; and if
equal to one another. The same, in tabular form.
(a) An isosceles triangle
To prove that (6) having the equal sides (a) The angles at the base produced.
are equal. (6) The angles on the
other side of the base
are equal. Particular Enunciation. Given.-The isosceles triangle (a) ABC, having
AB equal to AC, and the sides A B, AC
produced to D and E (Fig. 1). Required.-To prove that
(6) angle A B C is equal to angle A CB,
(c) angle D B C is equal to angle ECB. Construction.
In B D take any point F (Fig. 2). (d) From A E cut off a part A G' equal to AF
(by Euc. I. 3). Join B G and FC.
(The difficulty of this proposition arises from the fact thatthere are several triangles partly overlying each other.
Thus we have to deal with two pairs of triangles. The two triangles ABG, ACF: and the two triangles BCF, CDG. You will understand this at once if you
cut out four triangles in pasteboard of the same shape as those in the figure, and lay them over one another as they are there.)
(In Fig. 3 those triangles are shown separately, as they would appear if lifted off one another.)
In these triangles we have
A F equal to AG (Construction d),
are equal because, in Fig. 2, where they lie one
on the other, they coincide.) .. by Euc. I. 4, we have
(e) base F C equal to base BG. (f) angle AFC equal to angle AGB. (8) angle ACF equal to angle A BG. Now, we take triangles B CF, CB G (Fig. 2). (In Fig. 4 those triangles are shown separately, as
they would appear if lifted off one another.
A B is equal to AC (Given a). (h).. The remainder BF is equal to the
remainder CG (Axiom 3).
Then, in the triangles B CF, CB G we have
BF equal to C G (h),
by Euc. I. 4,
(k) angle FC B is equal to angle G B C. But it has been proved that
angle ABG is equal to angle A CF (8). (1) .. angle ABC is equal to angle ACB
(k and g). But, angles A B C, ACB (1) are the angles at the
base, and they are equal. And angles FBC, GCB (i) are the angles on the other side of the base, and they are equal.
Q. E. D. EXERCISE. - The pupil should now be asked to go through the proof, referring only to Fig. 2.
THEOREM (Euclid I. 6). Repeat.—The enunciation of Euc. I. 4 and
another, the sides which are opposite to
another. Particular Enunciation. Given.—The triangle ABC having (a) the
angle A B C equal to the angle ACB.
Required.–To prove that the side A B is equal
to the side A C. Construction.
Let us suppose that A B is not equal to A C;
Let us suppose that A B is greater than A C.
to AC. Join DC