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PROPOSITION XII. THEOREM.

69. CONVERSELY: When two straight lines are cut by a third straight line, if the alternate-interior angles be equal, the two straight lines are parallel.

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Let E F cut the straight lines A B and C D in the points H and K, and let the AHK = 2 HKD.

then

But

But

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Through the point H draw MN | to CD;

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.. A B, which coincides with MN, is to C D.

§ 68

Hyp.

Ax. 1.

Cons.

Q. E. D.

PROPOSITION XIII. THEOREM.

70. If two parallel lines be cut by a third straight line, the exterior-interior angles are equal.

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Let A B and C D be two parallel lines cut by the straight line E F, in the points H and K.

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71. COROLLARY. The alternate-exterior angles, EIIB and

C KF, and also A HE and DK F, are equal.

PROPOSITION XIV. THEOREM.

72. CONVERSELY: When two straight lines are cut by a third straight line, if the exterior-interior angles be equal, these two straight lines are parallel.

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Let EF cut the straight lines A B and C D in the points H and K, and let the EHB ZHKD.

We are to prove AB to CD.

=

Through the point H draw the straight line MN || to CD.

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.. A B, which coincides with MN, is to C D.

§ 70

Hyp.

Ax. 1.

Cons.

Q. E. D.

PROPOSITION XV. THEOREM.

73. If two parallel lines be cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles.

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Let A B and C D be two parallel lines cut by the straight line EF in the points H and K.

ZBHK+ Z HKD =two rt. .

then

We are to prove

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PROPOSITION XVI. THEOREM.

74. CONVERSELY: When two straight lines are cut by a third straight line, if the two interior angles on the same side of the secant line be together equal to two right angles, then the two straight lines are parallel.

[blocks in formation]

Let EF cut the straight lines A B and CD in the points H and K, and let the BHK + Z H K D equal two right angles.

Then

But

We

Te are to prove AB to CD.

Through the point H draw MN to CD.

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(being two interior on the same side of the secant line).

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§ 73

Нур.

:. LNH K+ZHKD=ZBHK+ZH KD. Ax. 1.

Take away from each of these equals the common ▲ II K D,

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.. A B, which coincides with MN, is to C D.

Cons.

Q. E D.

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