Question

select the correct answer
(overleftrightarrow{dg}) and (overleftrightarrow{eg}) are tangent to circle ( c ) and circle ( f ). the points of tangency are ( a ), ( b ), ( d ), and ( e ). if ( mangle dfe = 140^circ ), what is ( mangle acb )?

a. ( 160^circ )
b. ( 140^circ )
c. ( 120^circ )
d. ( 110^circ )

Explanation:

Step1: Identify tangent properties

Tangents from a point to a circle are equal, and radii to tangent points are perpendicular to tangents. So ∠FEG = ∠FDG = 90°, ∠CEG = ∠CAG = 90°.

Step2: Analyze quadrilateral FEGD

Sum of interior angles of quadrilateral is 360°. Given ∠DFE = 140°, ∠FEG = ∠FDG = 90°, so ∠EGD = 360° - 140° - 90° - 90° = 40°.

Step3: Analyze quadrilateral CEGA

∠CEG = ∠CAG = 90°, ∠EGA = ∠EGD = 40° (same angle). Sum of interior angles is 360°, so ∠ACB = 360° - 90° - 90° - 40° = 140°.

Answer:

B. 140°