| Given any triangle, extend its base to form l1, and draw l2 through the apex of the triangle, parallel to l1. Let q be the apex of the triangle, the anchor for l2. Three angles meet at q, and form the straight angle l2. Thus they sum to 180°. |
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Let ls be the left side of the triangle. The angle below l2 and to the left of ls is now alternate to, and congruent to, the angle above l1 and to the right of ls. (This was described in the previous section.) Thus the angle to the left of the apex is congruent to the lower left angle of the triangle. Similarly, the angle to the right of the apex is congruent to the lower right angle of the triangle. The three interior angles of the triangle are congruent to the three angles at q, which sum to 180°.
This theorem fails in spherical geometry. Start at the north pole and fly south to the equator. Turn 90° and fly west for a few hundred miles. Then turn 90° north and fly back to the north pole. This triangle has two right angles at the base and a nonzero angle at the apex, hence the angles sum to more than 180°.