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

4The

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.