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that is, the angle A = the angle C, and the angle B = theangle D.—(Euc. b. i, p. 15.)

The fum of the angles A + B, by the lad theo. isequal to a femicircle ; alfo the fum of the angles A + D,by the fame theo. is equal to a femicircle ; then A + B= A + D. Take away A from both fides of the equa-tion, and there will remain the angle B = the angle D.

’jth.—If a right line OR, fig. 12, cuts two parallel rightlines NP and S Qjjhe alternate angles NaR , QbO, are equal ,and confequently the lines parallel. —(Euc. b. 1, p. 29.)

The angles NaR and PaO, by the laft theo. are equal;and becaufe the lines are parallel A NaR is = A QbO== PaO, which by the laft theo. is = QbO.

8 th.—If any ftde of a right-lined triangle be continued, theexternal angle is equal to the fum of the two oppfite internalones. —(Euc. b. 1, p. 32.)— Fig. 13.

Let UST be the given triangle ; continue the fide UTto Z, and from T draw TY parallel to the fide US ; thenwill the external angle STZ be equal to the fum of theinternal angles SUT, UST ; for the lines SU and TYbeing parallel, tbe L STY is = A UST, by the lafttheo.; alfo the A ZTY is =r SUT, and confequentlythe A STZ is = A SUT + A UST, = the fum of theoppofite internal angles.

(jth.—The three angles of any rectangular triangle are to-gether equal to two right angles, or 18c degrees. —(Euc. b. 1,p. 32.)— Fig. 14.

In the triangle ABC, through A draw DE parallel tothe fide CB ; fo is the A DAC = the A ACB, by theo.7th ; as alfo the A E AB = tire A CB A -, but the ADAC + A CAB + A EAB = 180°, or two right an-

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