V. PARALLEL LINES 43 Figure 40 44. Theorem. The sum of the angles of a triangle is equal to two right angles. In triangle ABC (Fig. 40), we extend BC in the direction Cx, and draw CE parallel to AB. We form three angles at C (numbered 1 to 3 in the figure), whose sum is equal to two right angles. These angles are equal to the three angles of the triangle namely: 1 is the angle C of the triangle 2 = A (these are alternate interior angle formed by transversal AC with the parallel lines AB, CE) 3 = B (these are corresponding angles formed by transversal BC with the same parallels). Corollaries. I. An exterior angle of a triangle is equal to the sum of the non-adjacent interior angles. II. The acute angles of a right triangles are complementary. III. If two triangles have two pairs of equal angles, then the third pair of angles is equal as well. 44b. Theorem. The sum of the interior angles of a convex polygon4 is equal to two less than the number of sides times two right angles. Figure 41 Let the polygon be ABCDE (Fig. 41). Joining A to the other vertices, we decompose the polygon into triangles. The number of triangles is equal to the 4 The theorem remains true if one takes, as the interior angle at any vertex pointing inwards (Fig. 19b), the one which is directed towards the interior of the polygon, an angle which in this case is greater than two right angles. In the case of Fig. 19b the proof would proceed as in the text, drawing the diagonals starting from the vertex pointing inwards. If there are several inward-pointing vertices, the theorem would still be true, but the proof would be more diﬃcult.

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