THEOREM XXVII.

89. If two triangles have two sides of the one respectively equal to two sides of the other, and the included angles unequal, the third sides are unequal, and the greater third side is in the triangle having the greater included angle.

=

In the As ABC and DEF, let AC DF, CB = FE, and ACB > L c.

Place the

To prove that AB > DE.

DEF so that EF falls in its equal BC.

Let CH bisect L ECA, and draw EH.

Substitute AH and DE for their equals EH and EB.

Then

or

AH + HB> DE,

AB > DE.

Q. E. D.

THEOREM xxvIII.

90. CONVERSELY.-If two sides of a triangle are respectively equal to two sides of another, and the third sides unequal, the angle opposite the third side is greater in the triangle having the greater third side.

=

In the As ABC and DEF, let AC DF, BC=EF, and AB > DE.

But both these conclusions, being contrary to the hypothesis, are absurd.

... La cannot equal b, and cannot be less than

La> <b.

b.

Q. E. D.

THEOREM XXIX.

91. Two right-angled triangles are equal in all their parts if the hypothenuse and one side of the one are respectively equal to the hypothenuse and one side of the other.

In the RAS ABC and DEF, let the hypothenuse BC= EF, and AC = DF.

Place ABC on DEF so that AC falls in its equal DF.

THEOREM XXX.

92. Two right-angled triangles are equal in all their parts if the hypothenuse and one acute angle of the one are respectively equal to the hypothenuse and one acute angle of the other.

α=

In the RAS ABC and DEF, let BC = EF, and a = Lb.

93. COR.-Two right-angled triangles are equal if aside and an acute angle of the one are respectively equal to a side and an acute angle of the other.

RELATION BETWEEN THE PARTS

OF A TRIANGLE.

THEOREM XXXI.

94. In an isosceles triangle, the angles opposite the equal sides are equal.

In the isosceles ▲ ABC, let AC and BC be the equal sides.

95. COR. 1.-The straight line joining the vertex and the middle of the base of an isosceles triangle bisects the vertical angle and is perpendicular to the base.

96. COR. 2.-The straight line which bisects the vertical angle of an isosceles triangle bisects the base at right angles.

97. COR. 3.-Any equilateral triangle is equiangular.