Question

A circle is shown. Secants A B and B C intersects outside of the circle at point B. Secant A B intersects the circle at point D and secant B C intersects the circle at point E. The measure of arc D E is 25 degrees. The measure of arc A C is 110 degrees.
What is the measure of angle ABC?

42.5°
67.5°
85°
135°

Answer 1

Use the secant-secant angle theorem: an angle formed outside the circle by two secants equals half the difference of the measures of the intercepted arcs. Here the intercepted arcs are DE (25°) and AC (110°), so

m∠ABC = 1/2 |25° − 110°| = 1/2 · 85° = 42.5°.

Answer: 42.5°.