Question

dg and eg are tangent to circle c and circle f. the points of tangency are a, b, d, and e. if m∠dfe = 140°, what is m∠acb?
a. 160°
b. 140°
c. 120°
d. 110°

Explanation:

Step1: Recall the property of the sum of angles in a quadrilateral formed by tangents and radii

For circle \(F\), since \(DE\) and \(DG\) are tangents to circle \(F\), \(\angle FED=\angle FDG = 90^{\circ}\) (radius - tangent property). In quadrilateral \(FEDG\), the sum of interior angles is \(360^{\circ}\). Let \(\angle DFE = 140^{\circ}\), then \(\angle EGD=360^{\circ}-(90^{\circ}+90^{\circ}+140^{\circ}) = 40^{\circ}\)

Step2: Recall the property of the sum of angles in a quadrilateral formed by tangents and radii for circle \(C\)

For circle \(C\), since \(BA\) and \(BG\) are tangents to circle \(C\), \(\angle CAB=\angle CBG = 90^{\circ}\) (radius - tangent property). In quadrilateral \(CABG\), the sum of interior angles is \(360^{\circ}\). Let \(\angle AGB=\angle EGD = 40^{\circ}\) (common angle). Then \(\angle ACB=360^{\circ}-(90^{\circ}+90^{\circ}+40^{\circ})=140^{\circ}\)

Answer:

A. \(160^{\circ}\)