Question

what is the measure of minor arc \\(\overarc{eg}\\)? \\(17^{\circ}\\) \\(149^{\circ}\\) \\(75^{\circ}\\) \\(211^{\circ}\\) (and the circle diagram with center d, points e, f, g, h; angles \\(\angle edf = 66^{\circ}\\), \\(\angle fdg = 83^{\circ}\\))

Explanation:

Step1: Identify central angles for arc EG

The minor arc \( \widehat{EG} \) is formed by the central angles \( \angle EDG \). Wait, actually, looking at the diagram, the angles at center D: \( \angle EDF = 66^\circ \) and \( \angle FDG = 83^\circ \)? Wait no, wait. Wait, the angle between DE and DF is \( 66^\circ \), and between DF and DG is \( 83^\circ \)? Wait no, actually, to find the measure of minor arc EG, we need the central angle that subtends it. Wait, DE is a radius, DG is a radius. The angle between DE and DG: let's see, the angles around point D: the sum of angles around a point is \( 360^\circ \), but for minor arc, we need the smaller angle. Wait, no, wait the diagram: DE to DF is \( 66^\circ \), DF to DG is \( 83^\circ \)? Wait no, maybe I misread. Wait, the angle between DE and DG: let's add the two angles? Wait, no, wait the angle between DE (point E) and DG (point G): the angle at D between E and G. So DE to DF is \( 66^\circ \), DF to DG is \( 83^\circ \)? Wait, no, maybe the angle between DE and DG is \( 66^\circ + 83^\circ \)? Wait, let's check: if we have points E, F, G on the circle, with center D. So angle EDF is \( 66^\circ \), angle FDG is \( 83^\circ \). Then angle EDG (the central angle for arc EG) would be \( 66^\circ + 83^\circ = 149^\circ \)? Wait, but that seems like a major arc? Wait no, wait maybe I got the angles wrong. Wait, no, minor arc is the shorter arc between E and G. Wait, the total around the center is \( 360^\circ \), but if we have angle EDG as \( 66 + 83 = 149 \), then the other arc would be \( 360 - 149 = 211 \), which is major. So the minor arc EG is \( 149^\circ \)? Wait, but let's confirm. The central angle for a minor arc is the smaller angle between the two radii. Wait, but 149 vs 211, 149 is smaller? Wait no, 149 is less than 211? Wait 149 is less than 180? No, 149 is less than 180? Wait 180 is a semicircle. 149 is less than 180? Wait 149 is less than 180? Yes, 149 < 180. Wait, 66 + 83 is 149. So the central angle for arc EG is 149 degrees, so the minor arc EG is 149 degrees.

Step2: Calculate the central angle

So the central angle for arc EG is the sum of the two given angles: \( 66^\circ + 83^\circ = 149^\circ \). Since the measure of a minor arc is equal to the measure of its central angle (when it's the smaller arc), so the measure of minor arc \( \widehat{EG} \) is \( 149^\circ \).

Answer:

\( 149^\circ \) (corresponding to the option with \( 149^\circ \))