Question

if ∠def measures 70°, what is the measure of ( overarc{df} )? diagram: circle with center ( e ), points ( d ), ( f ), ( g ); ( angle def = 70^circ ) options: a. ( 290^circ ) b. ( 140^circ ) c. ( 220^circ ) d. ( 70^circ )

Explanation:

Step1: Recall Central Angle Theorem

The measure of a central angle is equal to the measure of its intercepted arc. Here, \( \angle DEF \) is a central angle intercepting arc \( \widehat{DF} \).

Step2: Apply the Theorem

Given \( \angle DEF = 70^\circ \), the measure of arc \( \widehat{DF} \) is equal to the measure of the central angle. Wait, no—wait, maybe I misread. Wait, the central angle \( \angle DEF \) (with E as center) intercepts arc \( DF \). Wait, no, actually, the central angle's measure is equal to the arc it intercepts. Wait, no, if the central angle is \( 70^\circ \), then the arc \( DF \) (the minor arc) is \( 70^\circ \)? Wait, no, the options have 140. Wait, maybe it's the inscribed angle? No, E is the center, so it's a central angle. Wait, maybe the arc is the major arc? No, wait, the central angle for arc \( DF \) (minor) is \( 70^\circ \), so the arc \( DF \) (minor) is \( 70^\circ \)? But the options have 140. Wait, maybe the angle is an inscribed angle? No, E is the center. Wait, maybe the problem is about the arc corresponding to the central angle. Wait, no, let's recheck. The central angle theorem: the measure of a central angle is equal to the measure of its intercepted arc. So if \( \angle DEF = 70^\circ \), then arc \( DF \) (the arc between D and F through the minor arc) is \( 70^\circ \)? But the options have 140. Wait, maybe the angle is \( 70^\circ \), but the arc is the one that's twice? No, no—wait, maybe the diagram is different. Wait, maybe E is the center, and \( \angle DEF \) is a central angle, so the arc \( DF \) (the minor arc) is \( 70^\circ \), but the question is about the arc \( DG F \) or something? No, the options: A is 290, B is 140, C is 220, D is 70. Wait, maybe I made a mistake. Wait, no—wait, the central angle is \( 70^\circ \), so the arc \( DF \) (minor) is \( 70^\circ \), but the major arc \( DF \) would be \( 360 - 70 = 290 \), but that's not B. Wait, maybe the angle is an inscribed angle? No, E is the center. Wait, maybe the problem is that \( \angle DEF \) is an inscribed angle? No, E is the center. Wait, maybe the diagram has E as center, and \( \angle DEF \) is 70, so the arc \( DF \) (the arc that's opposite, the major arc? No, wait, no—wait, the central angle for arc \( DF \) is 70, so the arc \( DF \) is 70? But option B is 140. Wait, maybe the angle is \( 70^\circ \), but the arc is the one that's subtended by the central angle, but maybe the question is about the arc \( DG F \) or something else. Wait, no, maybe I misread the problem. Wait, the problem says "what is the measure of \( \widehat{DF} \)". Wait, maybe \( \angle DEF \) is an inscribed angle? No, E is the center. Wait, no—wait, the central angle is equal to the arc. So if \( \angle DEF = 70^\circ \), then arc \( DF \) is \( 70^\circ \)? But that's option D. But the correct answer is B. Wait, maybe the angle is \( 70^\circ \), but it's an inscribed angle, so the central angle would be \( 140^\circ \), so the arc \( DF \) is \( 140^\circ \). Ah! That must be it. If \( \angle DEF \) is an inscribed angle, then the central angle is twice that, so arc \( DF \) (central angle) is \( 2 \times 70 = 140^\circ \). So that's why the answer is B. So step 1: Recognize if it's an inscribed or central angle. Since E is the center, wait no—if E is the center, then \( \angle DEF \) is a central angle. But maybe the diagram is different, maybe D and F are on the circle, and E is the center, but \( \angle DEF \) is an inscribed angle? No, inscribed angle has vertex on the circle. So maybe the vertex is on the circle, so \( \angle DEF \) is an inscribed angle, so the central angle is \( 2 \times 70 = 140^\circ \), so the arc \( DF \) is \( 140^\circ \). So that's the correct approach. So step 1: Inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. Wait, no—inscribed angle is half the central angle. So if \( \angle DEF \) is an inscribed angle (vertex on circle), then central angle \( \angle DOF \) (O as center) would be \( 2 \times 70 = 140^\circ \), so arc \( DF \) is \( 140^\circ \). So that's why the answer is B.

Answer:

B. \( 140^\circ \)