In the diagrams below, the points *A* and *B* are located anywhere along the rays forming ∠*C*, whose measure is always 40°. The points *A* and *B* are connected by a line segment to form △*ABC*. The point *D* is the intersection of the bisectors of the exterior angles at *A* and *B*.

What is the measure of ∠

*ADB*?By the exterior angle theorem:

2

2

∠1 + ∠2 = 2

*x*= ∠2 + 40 → ∠2 = 2*x*– 402

*y*= ∠1 + 40 → ∠1 = 2*y*– 40∠1 + ∠2 = 2

*x*+ 2*y*– 80But ∠1 + ∠2 = 180 – 40 = 140

140 = 2

2

*x*+ 2*y*– 802

*x*+ 2*y*= 220 →*x*+*y*= 110*z*= 180 – (*x*+*y*) = 180 – 110 = 70So the measure of ∠*ADB* = *z* is 70°, no matter where *A* and *B* are placed.