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By David Gans

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Before proving them, let us agree that in a Saccheri quadrilateral ABCD (Fig. Ill, 2), with right angles at A and B, AB will be called the base, CD the summit, and AD, BC the arms. Theorem 29. The summit angles of a Saccheri quadrilateral are equal. Proof. Consider the Saccheri quadrilateral ABCD (Fig. Ill, 2). Triangles ABC, BAD are congruent by side-angle-side (Theo. 4). Hence AC = BD. * Since Euclid proved Theorems 4 and 8 by superposition, we are regarding the facts they state as assumptions (see §2, Property 13).

Show that if a quadrilateral with a base has equal summit angles relative to the base, it will have equal arms relative to the base and hence be a Saccheri quadrilateral. 6. Prove that two quadrilaterals are congruent if a side, angle, side, angle, side of one are equal to the corresponding parts of the other, the five parts in each case being consecutive on the quadrilateral. 7. * 4. THE HYPERBOLIC PARALLEL POSTULATE There are many choices for the statement to serve as the hyperbolic parallel postulate.

43), we could say that there are triangles with arbitrarily small defects. The term "defect," however, is no mere verbal convenience, but represents an important idea, as will be seen when we study the area of a triangle. In order to exhibit a basic fact about defects which is relevant to that study, let us now take any triangle ABC (Fig. Ill, 18) and consider the two A D B Fig. Ill, 18 triangles A DC and BDC formed by joining C to any point D between A and B. t * Two quadrilaterals are, by definition, congruent if their sides and angles are equal, respectively.

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