What is the measure of ∠ADB\angle ADB∠ADB? It will be the same regardless of the placement of points AAA and BBB, as the figure below suggests, as long as the indicated angles are equal. Tags: Problem Set 5,Geometry,Geometry-Euclidean Solution By the exterior angle theorem (twice):2x=∠2+40°2y=∠1+40°.\begin{aligned}2x &= \angle 2 + 40° \\2y &= \angle 1 + 40°.\end{aligned}2x2y=∠2+40°=∠1+40°.Therefore,∠2=2x−40∠1=2y−40.\begin{aligned}\angle 2 &= 2x - 40 \\\angle 1 &= 2y - 40.\end{aligned}∠2∠1=2x−40=2y−40.Adding gives ∠1+∠2=2x+2y−80\angle 1 + \angle 2 = 2x + 2y -80∠1+∠2=2x+2y−80, but2x+2y−80=∠1+∠2=180−40=140,\begin{aligned}2x + 2y - 80 &= \angle 1 + \angle 2 \\&= 180 - 40 \\&= 140,\end{aligned}2x+2y−80=∠1+∠2=180−40=140,so 2x+2y=2202x + 2y = 2202x+2y=220, or x+y=110x + y = 110x+y=110. Finally, z=180−(x+y)=180−110=70°z = 180 - (x + y) = 180 - 110 = 70°z=180−(x+y)=180−110=70°.
