The First And Second Books Of Euclid, Explained To Beginners -

Summary: This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1872 Excerpt: ...and a side of the first A equal to a side of the second, similarly placed with respect to the equal Zs, those As are also equal in every other respect. (Prop. XXVI.) 4. That if two lines are parallel and a third line meets them, the alternate Zs so formed are equal. (Prop. XXIX.) Let A B D ...show moreC be a parallelogram, that is, suppose it to be a four-sided figure, whose A B opposite sides are parallel, and let A D Vbe one of its diagonals. We have to prove that the opposite sides are equal: --namely, the side D A B to the side C D, and the side A C to the side B D: and also that the opposite Zs are equal, namely, the / ABD to the Z ACD, and the Z CAB to the Z BDC and that the parallelogram is divided into two equal parts by its diagonal A D. Proof. Because A B and C D are parallel, and the line AD meets them, it follows that the alternate Zs so formed, namely, BAD and ADC, are equal. (Prop. XXIX.) Because A C and B D are parallel and the line A D meets them, it follows (Prop. XXIX.) that the alternate Zs s0 formed, namely, CAD and A D B, are equal. Hence, the two As ABD and CD A have the two Zs B AD and B D A of the one equal respectively to the two /s ADC and C AD of the other, and the side AD common to the two As and similarly placed with respect to the equal Z s It follows, therefore (Prop. XXVI.), that these As are also equal in every other respect. That is to say, the Z ABD is equal to the / ACD; the side AB to the side C D, the side B D to the side A C, and the area of the one A to the area of the other. Moreover, since it was shown that the two parts BAD and CAD of the Z. B AC are equal respectively to the two parts, ADCandBDA, of the Z BDC, if the one pair of equal be added to the other the sums will be equal; that is to say, the /_ B A C is equal to the Z... ...show less