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So, if the interior angles on the same side of the transversal are not supplemental, the two lines meet; and as, by 143, they cannot meet on that side of the transversal where the two interior angles are greater than a straight angle, therefore they must meet on the side where the two interior angles are together less than a straight angle.

172. CONTRANOMINAL OF 99. Lines in the same plane parallel to the same line cannot intersect, and so are parallel to one another.

THEOREM XXI.

173. Each exterior angle of a triangle is equal to the sum of the two interior opposite angles.

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HYPOTHESIS. ABC any A, with side AB produced to D.
CONCLUSION. CBD = A + 4C.

PROOF. From B, by 167, draw BF || AC. Then, by 168, 40 = * CBF, and by 169, 4 A = * DBF;

by adding, X A + 4C = FBD + X CBF = X CBD.

EXERCISES. 27. Each angle of an equilateral triangle is two-thirds of a right angle. Hence show how to trisect a right angle.

28. If any of the angles of an isosceles triangle be twothirds of a right angle, the triangle must be equilateral.

THEOREM XXII.

174. The sum of the three interior angles of a triangle is equal to a straight angle.

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HYPOTHESIS. ABC any A.
CONCLUSION. CAB + x B + XC = st. 4.
PROOF. Produce a side, as CA, to D.
By 173, B+C = X DAB.
Add to both sides X BAC.

...

* CAB + B + C = DAB + % BAC = st. 4.

175. COROLLARY.

In any right-angled triangle the two acute angles are complemental.

CHAPTER VII.

TRIANGLES.

THEOREM XXIII.

176. Two triangles are congruent if two angles and an opposite side in the one are equal respectively to two angles and the corresponding side in the other.

A A

HYPOTHESIS. ABC and DFG AS, with

XA = = XD,
XC = XG,
AB = DF

. CONCLUSION. A ABCEA DFG.

PROOF. By 174, 4A + B + $C = st. * = 4D + F + *G, By hypothesis, % A + 4.C = 4D + G,

.. XB

X B = F,
(89. If equals be taken from equals, the remainders are equal.)

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(128. Triangles are congruent if two angles and the included side are equal in each.) THEOREM XXIV.

177. If two triangles have two sides of the one equal respectively to two sides of the other, and the angles opposite to one pair of equal sides equal, then the angles opposite to the other pair of equal sides are either equal or supplemental.

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The angles included by the equal sides must be either equal or unequal. CASE I. If they are equal, the third angles are equal.

(174. The sum of the angles of a triangle is a straight angle.) CASE II. If the angles included by the equal sides are unequal, one must be the greater.

HYPOTHESIS. ABC and FGH AS, with

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CONCLUSION. XC+*H = st. 4.
PROOF. By 164, make the % ABD XG

Δ ABD2 Δ FGH, (128. Triangles are congruent if two angles and the included side are equal in each.)

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BD,

But, by hypothesis, BC = GH,

.. BC

.:. * BDC = 4C.
(126. In an isosceles triangle the angles opposite equal sides are equal.)

But % BDA + XBDC = st. %,

XH + XC = st. 4.

178. COROLLARY I. If two triangles have two sides of the one respectively equal to two sides of the other, and the angles opposite to one pair of equal sides equal, then, if one of the angles opposite the other pair of equal sides is a right angle, or if they are oblique but not supplemental, or if the side opposite the given angle is not less than the other given side, the triangles are congruent.

179. COROLLARY II. Two right-angled triangles are congruent if the hypothenuse and one side of the one are equal respectively to the hypothenuse and one side of the other.

On the Conditions of Congruence of Two Triangles.

180. A triangle has three sides and three angles.

The three angles are not all independent, since, whenever two of them are given, the third may be determined by taking their sum from a straight angle.

In four cases we have proved, that, if three independent parts of a triangle are given, the other parts are determined; in other words, that there is only one triangle having those

parts:

(124) Two sides and the angle between them.
(128) Two angles and the side between them.
(129) The three sides.
(176) Two angles and the side opposite one of them.
In the only other case, two sides and the angle opposite one

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