In the diagram at the right, <CBD is 3 times as great as <DBE. What is the measure of <DBA? Explain your reasoning.
We are given ray BC is perpendicular to line AE. Ray BD is in the interior of angle CBE.
Angle CBD is adjacent to angle DBE; so we know angles CBD and DBE are complementary. (If the exterior sides of two adjacent angles are perpendicular then the angles are complementary.)
Let the measure of angle DBE be x. Then the measure of angle CBD is 3x (since its measure is 3 times the other.)
Since the angles are complementary we have 3x+x=90 or x=22.5. The measure of angle CBD is 3(22.5)=67.5
The measure of angle DBA is the measure of angle DBC plus the measure of angle CBA by the angle addition postulate. Further, from the definition of perpendicular we know that the measure of angle CBA is 90.
Thus the measure of angle DBA is 67.5+90=157.5