Question

which arc is congruent to \\(\overarc{eh}\\)?
\\(\bigcirc\\) \\(\overarc{gh}\\)
\\(\bigcirc\\) \\(\overarc{fh}\\)
\\(\bigcirc\\) \\(\overarc{ge}\\)
\\(\bigcirc\\) \\(\overarc{fg}\\)
(there is a circle with center d. points e, f, g, h are on the circle. the angle \\(\angle edf\\) is 55°, \\(\angle fdg\\) is 70°, \\(\angle gdh\\) is 110°.)

Explanation:

Step1: Find measure of arc EH

The central angle for arc \( \widehat{EH} \) is the sum of the angles \( \angle EDF = 55^\circ \) and the straight line? Wait, no, looking at the diagram, the central angle for \( \widehat{EH} \): Wait, actually, let's calculate the measure of arc \( \widehat{EH} \). The total around a point is \( 360^\circ \), but maybe we can find the central angle. Wait, the given angles: \( \angle EDF = 55^\circ \), \( \angle FDG = 70^\circ \), \( \angle GDH = 110^\circ \). Then the remaining angle \( \angle HDE \): Wait, no, let's check the arcs. Wait, arc \( \widehat{EH} \): Let's find its measure. Wait, maybe first, let's find the measure of arc \( \widehat{EH} \). Wait, the central angle for arc \( \widehat{EH} \): Let's see, the sum of angles around D is \( 360^\circ \). So \( 55^\circ + 70^\circ + 110^\circ + \text{angle for } \widehat{EH} \)? No, maybe I made a mistake. Wait, actually, the diagram: points E, F, G, H on the circle, center D. So \( \angle EDF = 55^\circ \), \( \angle FDG = 70^\circ \), \( \angle GDH = 110^\circ \). Then the angle \( \angle HDE \) would be \( 360 - 55 - 70 - 110 = 125^\circ \)? No, that can't be. Wait, maybe the arcs: Wait, no, maybe the arc \( \widehat{EH} \) is actually the sum of \( \angle EDF + \angle FDG + \angle GDH \)? No, that doesn't make sense. Wait, maybe I misread. Wait, the problem is to find which arc is congruent to \( \widehat{EH} \). Congruent arcs have equal measures (since they are in the same circle, same radius). So first, find the measure of \( \widehat{EH} \). Wait, maybe the central angle for \( \widehat{EH} \): Wait, looking at the options, let's check each arc's measure.

First, let's find the measure of \( \widehat{EH} \). Wait, maybe the arc \( \widehat{EH} \) is formed by the central angle that is \( 55^\circ + 70^\circ + 110^\circ \)? No, that's too big. Wait, no, maybe the diagram is a circle with center D, and the arcs: \( \widehat{FG} \) has central angle \( 70^\circ \), \( \widehat{GE} \): Wait, no, let's re-express. Wait, the angle \( \angle EDF = 55^\circ \), so arc \( \widehat{EF} \) is \( 55^\circ \). Arc \( \widehat{FG} \) is \( 70^\circ \) (central angle \( 70^\circ \)). Arc \( \widehat{GH} \) is \( 110^\circ \) (central angle \( 110^\circ \)). Then the remaining arc \( \widehat{HE} \) would be \( 360 - 55 - 70 - 110 = 125^\circ \)? No, that can't be. Wait, maybe I'm wrong. Wait, the question is which arc is congruent to \( \widehat{EH} \). Let's check the options:

Option 1: \( \widehat{GH} \): measure \( 110^\circ \)

Option 2: \( \widehat{FH} \): \( \widehat{FH} \) is a straight line? Wait, no, \( \widehat{FH} \) would be \( 55 + 70 + 110 = 235^\circ \)? No, that's not. Wait, maybe \( \widehat{FH} \) is a diameter? No, the central angle for \( \widehat{FH} \): Wait, \( \angle FDH \): \( 55 + 70 + 110 = 235 \)? No, that's not. Wait, maybe the arc \( \widehat{EH} \) has measure equal to \( \angle EDH \). Wait, maybe I made a mistake. Wait, let's calculate the measure of \( \widehat{EH} \). Wait, the sum of angles around D: \( 55^\circ + 70^\circ + 110^\circ + x = 360^\circ \), so \( x = 360 - 55 - 70 - 110 = 125^\circ \). No, that's not matching. Wait, maybe the diagram is different. Wait, maybe the arc \( \widehat{EH} \) is actually the arc from E to H, passing through... Wait, no, maybe the central angle for \( \widehat{EH} \) is \( 55^\circ + 70^\circ = 125^\circ \)? No, \( 55 + 70 = 125 \), then \( 125 + 110 = 235 \), no. Wait, maybe the arc \( \widehat{EH} \) is \( 180 - 55 = 125 \)? No. Wait, maybe the correct approach is: congruent arcs have equal measures. Let's check each option:

Arc \( \widehat{GH} \): measure \( 110^\circ \) (central angle \( 110^\circ \))

Arc \( \widehat{FH} \): Let's see, \( \widehat{FH} \) would be from F to H. The central angle for \( \widehat{FH} \) is \( 55^\circ + 70^\circ + 110^\circ = 235^\circ \)? No, that's not. Wait, maybe \( \widehat{FH} \) is a major arc. No, that's not. Wait, arc \( \widehat{GE} \): Let's find its measure. \( \widehat{GE} \) would be from G to E. The central angle: \( 70^\circ + 55^\circ = 125^\circ \)? No, \( 70 + 55 = 125 \), then plus \( 110 \)? No. Wait, arc \( \widehat{FG} \): measure \( 70^\circ \). Wait, maybe I made a mistake. Wait, the problem is to find which arc is congruent to \( \widehat{EH} \). Let's recalculate the measure of \( \widehat{EH} \). Wait, the total around the center is \( 360^\circ \). So the arcs: \( \widehat{EF} = 55^\circ \), \( \widehat{FG} = 70^\circ \), \( \widehat{GH} = 110^\circ \), so \( \widehat{HE} = 360 - 55 - 70 - 110 = 125^\circ \). Now check the options:

- \( \widehat{GH} \): \( 110^\circ \) → not equal to \( 125^\circ \)
- \( \widehat{FH} \): \( \widehat{FH} \) is \( \widehat{EF} + \widehat{FG} + \widehat{GH} = 55 + 70 + 110 = 235^\circ \) → no
- \( \widehat{GE} \): \( \widehat{GE} \) is \( \widehat{FG} + \widehat{EF} = 70 + 55 = 125^\circ \)? Wait, no, \( \widehat{GE} \) is from G to E. So the central angle would be \( 360 - 110 = 250^\circ \)? No, that's not. Wait, maybe I messed up the direction. Wait, the arc \( \widehat{EH} \): let's see, the central angle for \( \widehat{EH} \) is \( \angle EDH \). Let's calculate \( \angle EDH \): \( 360 - 55 - 70 - 110 = 125^\circ \). Now, arc \( \widehat{GE} \): what's its measure? Wait, \( \angle GDE \): \( \angle GDF + \angle FDE = 70 + 55 = 125^\circ \). Oh! So the central angle for \( \widehat{GE} \) is \( 125^\circ \), same as \( \widehat{EH} \). Wait, no, wait: \( \angle GDE = 70 + 55 = 125^\circ \), so arc \( \widehat{GE} \) has measure \( 125^\circ \), same as \( \widehat{EH} \). Wait, but the options: one of the options is \( \widehat{GE} \)? Wait, the options are \( \widehat{GH} \), \( \widehat{FH} \), \( \widehat{GE} \), \( \widehat{FG} \). Wait, let's check again.

Wait, maybe the arc \( \widehat{EH} \) is actually the sum of \( \angle EDF = 55^\circ \) and the angle opposite? No, wait, let's re-express:

Wait, the given angles: \( \angle EDF = 55^\circ \), \( \angle FDG = 70^\circ \), \( \angle GDH = 110^\circ \). Then the angle \( \angle HDE \) is \( 360 - 55 - 70 - 110 = 125^\circ \). So arc \( \widehat{EH} \) has measure \( 125^\circ \) (since central angle is \( 125^\circ \)). Now, arc \( \widehat{GE} \): the central angle is \( \angle GDE \), which is \( \angle GDF + \angle FDE = 70 + 55 = 125^\circ \). So arc \( \widehat{GE} \) has measure \( 125^\circ \), same as \( \widehat{EH} \). Wait, but let's check the other options:

- \( \widehat{GH} \): \( 110^\circ \) (central angle \( 110^\circ \)) → not equal.
- \( \widehat{FH} \): central angle \( 55 + 70 + 110 = 235^\circ \) → no.
- \( \widehat{FG} \): \( 70^\circ \) → no.
- \( \widehat{GE} \): central angle \( 125^\circ \) → same as \( \widehat{EH} \). So they are congruent.

Wait, but let's confirm. So arc \( \widehat{EH} \) has measure \( 125^\circ \), arc \( \widehat{GE} \) has measure \( 125^\circ \), so they are congruent.

Step2: Compare with options

So the arc congruent to \( \widehat{EH} \) is \( \widehat{GE} \).

Answer:

\( \boldsymbol{\widehat{GE}} \) (the option with \( \widehat{GE} \))