PROPOSITION XXIV. THEOREM.

107. Two triangles are equal in all respects when a side and two adjacent angles of the one are equal respectively to a side and two adjacent angles of the other.

C

BA'

In the triangles ABC and A'B'C', let A B

=

ZA ZA', ▲ B = Z B'.

We are to prove

AABC = A A'B' C.

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

AC will take the direction of A' C',

(for LA LA', by hyp.) ;

the point C, the extremity of A C, will fall upon A'C' or A'C' produced.

BC will take the direction of B' C',

(for LB = LB', by hyp.) ;

the point C, the extremity of BC, will fall upon B'C' or B'C' produced.

.. the point C, falling upon both the lines A'C' and B' C', must fall upon a point common to the two lines, namely, C'.

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

Q. E. D.

108. Two triangles are equal when the three sides of the one are equal respectively to the three sides of the other.

In the triangles A B C and A'B'C', let A B = A' B', A CA' C', BC= B'C'.

Place A A'B'C' in the position A B' C, having its greatest side A'C' in coincidence with its equal A C, and its vertex at B', opposite B.

Draw B B' intersecting A C at H.

point A is at equal distances from B and B'.

point C is at equal distances from B and B'.

.. AC is to B B' at its middle point,

Нур.

Нур.

§ 60

(two points at equal distances from the extremities of a straight line determine the at the middle of that line).

Now if AA B' C be folded over on AC as an axis until it comes into the plane of ▲ ABC,

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

Q. E. D.

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.

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

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= L A' 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 cut off equal distances from the

foot of the 1).

.. the two 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 A C = A' C',

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.

..A ACEA BCE,

(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).

Q. E. D.

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.