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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.

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Let the line C E bisect the ACB of the isosceles

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

Also, II.

and the included of the other).

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

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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.

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In the triangle ABC, let the ▲ B=LC.

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

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Ex. Show that an equiangular triangle is also equilateral.

PROPOSITION XXXI. THEOREM.

115. If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first will be greater than the third side of the second.

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In the ▲ A B C and ABE, let A B = A B, BC=BE; ABC > ZA BE.

but

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Place the

so that A B of the one shall coincide with A B

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

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of the other).

and the included
.. EF FC,

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(being homologous sides of equal ▲).

AFFE> A E,

(the sum of two sides of a ▲ is greater than the third side).

Substitute for FE its equal FC. Then

AFFC > A E; or,

AC > AE.

$ 96

Q. E. D.

PROPOSITION XXXII. THEOREM.

116. CONVERSELY: If two sides of a triangle be equal respectively to two sides of another, but the third side of the first triangle be greater than the third side of the second, then the angle opposite the third side of the first triangle is greater than the angle opposite the third side of the second.

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In the AABC and A'B'C', let A B = A' B', A CA' C';

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(having two sides and the included

and

And if

then would

= Δ Α' Β' Γ',

§ 106

of the one equal respectively to two sides

and the included of the other),

BCB'C',

(being homologous sides of equal ▲).

A < A',

BCB'C',

$ 115

(if two sides of a ▲ be equal respectively to two sides of another A, but the included of the first be greater than the included of the second, the third side of the first will be greater than the third side of the second.) But both these conclusions are contrary to the hypothesis ; .. A does not equal A', and is not less than A'. .LA > Z A'.

Q. E. D

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117. Of two sides of a triangle, that is the greater which is opposite the greater angle.

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In the triangle ABC let angle ACB be greater than

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(the sum of two sides of a ▲ is greater than the third side).

Substitute for EC its equal E B. Then

AE+EB> A C, or

AB A C.

$ 114

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Q. E. D.

Ex. ABC and ABD are two triangles on the same base A B, and on the same side of it, the vertex of each triangle being without the other. If AC equal A D, show that BC cannot equal B D.

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