...contradict the hypothesis that angle «> angle c. Therefore AB<BC. QED PROPOSITION XXII. THEOREM 79. If two triangles have two sides and the included angle...making AB coincide with its equal A'B' Then, since a=a', the side AC will fall along A'C'. Also, since AC=A'C', the point C will fall on C'. Then BC will...

...are unequal, the opposite angles are unequal, and the greater angle is opposite the greater side.] PROPOSITION XXII. THEOREM 70. If two triangles have...included angle of the other, the triangles are equal. GIVEN—AB, AC, and a, of the triangle ABC respectively equal to A'B', A'C, and a', of the triangle...

...equal, the figure is a rectangle. 46. Prove by coincidence that two parallelograms are equal if they have two sides and the included angle of one equal respectively to two sides and the included angle of the other. 42 PLANE GEOMETRY PROPOSITION XXVIII 109. Theorem. // two sides of a quadrilateral are...

...on a fixed great circle. 26. If two spherical triangles have two sides and the included angle of the one equal respectively to two sides and the included angle of the other, the two triangles are equal or symmetrical. 27. If two convex spherical triangles have two sides of the...

...Proposition XIX.] GIVEN in the triangle ABC, the angle a >angle c. To PROVE side BC> side AB. Either AB is equal to, greater than, or less than BC. If...equal respectively to two sides and the included angle of the othcr, the triangles are equal. C' GIVEN — AB, AC, and a, of the triangle ABC respectively...

...contradict the hypothesis that angle «> angle c. Therefore AB>BC QED PROPOSITION XXII. THEOREM 78. If two triangles have two sides and the included angle...and the included angle of the other, the triangles arc equal. JB' C' GIVEN — AB, AC, and a, of the triangle ABC respectively equal to A'B', A'C, and...

...PROBLEM. To construct a square on a given side. PROPOSITION XXVI. THEOREM. 115. If two parallelograms have two sides and the included angle of one equal...two sides and the included angle of the other, the parallelograms are equal. 7 ».D *a FIG. 53. [Superpose, and use Ax. III.] 116. COR. If two rectangles...

...I. (MODEL FOR PUPILS.) /. Theorem. If any two triangles have two sides and the included angle of the one equal respectively to two sides and the included angle of the other, the triangles are equal. //. Given: Д ABC and Л DEF, in which (1) AB = DE, (2) A С = D К, and (3) z BA С = ZED F. Required:...

...§ 600.) PROP. XXII. THEOREM. 602. If two spherical triangles on the same sphere, or equal spheres, have two sides and the included angle of one equal,...respectively, to two sides and the included angle of the other, II. They are symmetrical if the equal parts occur in the reverse order. I. Given ABC...

...§ 600.) PROP. XXII. THEOREM. 602. If two spherical triangles on the same sphere, or equal spheres, have two sides and the included angle of one equal,...respectively, to two sides and the included angle of the other, I. They are equal if the equal parts occur in the same order. II. They are symmetrical...