The following diagram shows a triangle with each of its sides produced both ways : What is the sum of degree measures of the angles numbered ?

When each side of a triangle is produced both ways, two exterior angles are formed at each of the three vertices, totaling six exterior angles [t3]. At any single vertex, an interior angle (let's call it 'a') and its adjacent exterior angle are supplementary, summing to 180° [t1][t2]. Since each side is extended in both directions, there are two such exterior angles at each vertex, each equal to (180° - interior angle) [t3]. For a triangle with interior angles A, B, and C, the sum of these six exterior angles is: (180 - A) + (180 - A) + (180 - B) + (180 - B) + (180 - C) + (180 - C). This simplifies to 1080° - 2(A + B + C). Since the sum of interior angles (A + B + C) is 180°, the total sum is 1080° - 360° = 720° [t3][t4].