109. Two right triangles are equal when a side and the hypotenuse of the one are equal respectively to a side and the hypotenuse of the other.

A

Α'

B

C B

C

In the right triangles A B C and A' B'C', let A B= A' B',

and AC = A' C'.

We are to prove

AABC = A A'B'C'.

Take up the ▲ A B C and place it upon ▲ A'B'C', so that A B will coincide with A' B'.

Then

BC will fall upon B'C',

(for LABC= LA'B'C', each being a rt. 4),

and point C will fall upon C";

otherwise the equal oblique lines A C and A'C' would cut off unequal distances from the foot of the L, which is impossible,

$ 57 (two equal oblique lines from a point in a 1 cut off equal distances from the

foot of the 1).

.. the two A coincide, and are equal in all respects.

Q. E. D.

110. Two right triangles are equal when the hypotenuse and an acute angle of the one are equal respectively to the hypotenuse and an acute angle of the other..

In the right triangles A B C and A' B' C', let AC = A' C',

(if two rt. A have an acute ▲ of the one equal to an acute then the other acute & are equal).

of the other,

.. ΔΑΒΟ

=

A A'B'C',

$107

(two are equal when a side and two adj.

respectively to a side and two adj.

111. COROLLARY. Two right triangles are equal when a side and an acute angle of the one are equal respectively to an homologous side and acute angle of the other.

of the one are equal

of the other).

Q. E. D.

PROPOSITION XXVIII. THEOREM.

112. In an isosceles triangle the angles opposite the equal sides are equal.

Let ABC be an isosceles triangle, having the sides AC and CB equal.

From C draw the straight line CE so as to bisect the ZACB.

=

..AACE ABCE,

(two are equal when two sides and the included

respectively to two sides and the included

Hyp.

Iden.

Cons.

§ 106

of the one are equal

of the other).

Ex. If the equal sides of an isosceles triangle be produced, show that the angles formed with the base by the sides produced are equal.

113. A straight line which bisects the angle at the vertex af an isosceles triangle divides the triangle into two equal triangles, is perpendicular to the base, and bisects the base.

Let the line C E bisect the ACB of the isosceles

(having two sides and the included of the one equal respectively to two sides

Also, II.

and the included of the other).

(a straight line meeting another, making the adjacent & equal, is to

PROPOSITION XXX. THEOREM.

114. If two angles of a triangle be equal, the sides opposite the equal angles are equal, and the triangle is isosceles.

In the triangle ABC, let the B = L C.

(having a side and an acute ▲ of the one equal respectively to a side and an

acute of the other).

=

.. AB AC,

(being homologous sides of equal ▲).

Q. E. D.

Ex. Show that an equiangular triangle is also equilateral.