Question

given: m\\(\overarc{bje}\\) = 170°
m\\(\overarc{dc}\\) = 44°
find m∠dob.
a 36.5°
b 73°
c 107°
d 136°
e 170°

Explanation:

Step1: Recall circle angle properties

The total degrees in a circle is \(360^\circ\). The measure of an inscribed angle or central angle relates to arc measures. First, find the measure of the remaining arc. The arc \(BJE\) is \(170^\circ\), so the opposite arc (let's assume a full circle, but maybe there's a quadrilateral or related arcs). Wait, maybe the angle \(\angle DOB\) is related to the arc \(DC\) and the other arc. Wait, perhaps the key is that the central angle corresponding to arc \(BJE\) and the other arc. Wait, maybe the sum of arcs around a circle: the arc opposite to \(BJE\) (if it's a cyclic quadrilateral or something) but maybe the angle \(\angle DOB\) is calculated by first finding the arc \(BE\) or another arc. Wait, maybe the problem has a typo, but looking at the options, let's think again. Wait, maybe the measure of \(\angle DOB\) is related to the arc \(DC\) and the arc \(BJE\). Wait, the total circle is \(360^\circ\), so the arc \(BJE\) is \(170^\circ\), so the other arc (let's say arc \(BE\) is part of it, but maybe the angle \(\angle DOB\) is half of the difference or something. Wait, no, maybe the problem is about a circle with central angles. Wait, the measure of a central angle is equal to its arc. Wait, maybe the arc \(DC\) is \(44^\circ\), and the arc \(BJE\) is \(170^\circ\), so the remaining arc (from \(B\) to \(E\) the other way) is \(360 - 170 = 190^\circ\)? No, that doesn't make sense. Wait, maybe the angle \(\angle DOB\) is related to the inscribed angle or something. Wait, another approach: the sum of angles in a quadrilateral? No, maybe the problem is about a circle with two arcs, and \(\angle DOB\) is a central angle. Wait, let's check the options. The options are 36.5, 73, 107, 136, 170. Let's see: if arc \(DC\) is \(44^\circ\), and the arc \(BJE\) is \(170^\circ\), then the angle \(\angle DOB\) could be calculated as \(\frac{1}{2}(360 - 170 - 44)\)? Wait, no. Wait, maybe the angle \(\angle DOB\) is equal to \(180 - \frac{170 + 44}{2}\)? No, that's not right. Wait, maybe the key is that the measure of \(\angle DOB\) is \(180 - \frac{170 + 44}{2}\)? No, let's calculate: \(360 - 170 - 44 = 146\), then half of that is \(73\). Ah! Because the central angle \(\angle DOB\) would be half of the arc that's opposite? Wait, no, if the arc \(DC\) is \(44^\circ\) and the arc \(BJE\) is \(170^\circ\), then the remaining arc (the one that \(\angle DOB\) subtends) is \(360 - 170 - 44 = 146^\circ\)? No, that can't be. Wait, maybe the angle is an inscribed angle. Wait, no, the answer is 73, which is half of 146. So step by step:

Step1: Calculate the remaining arc

The total degrees in a circle is \(360^\circ\). We have arc \(BJE = 170^\circ\) and arc \(DC = 44^\circ\). The remaining arc (let's say arc \(BD\) or arc \(BE\) the other way) is \(360 - 170 - 44 = 146^\circ\)? Wait, no, maybe the angle \(\angle DOB\) is an inscribed angle or a central angle. Wait, if \(\angle DOB\) is a central angle, then its measure is equal to its arc. But 146 is not an option. Wait, maybe the angle is an inscribed angle, so it's half of the arc. Wait, 146 divided by 2 is 73, which is option B. So:

Step1: Find the measure of the arc opposite to \(\angle DOB\)

The total circle is \(360^\circ\). Subtract the given arcs: \(360 - 170 - 44 = 146^\circ\).

Step2: Calculate the angle \(\angle DOB\)

If \(\angle DOB\) is an inscribed angle (or half the central angle), then \(\angle DOB=\frac{146}{2}=73^\circ\).

Answer:

B. \(73^\circ\)