Question

use the drop - down menus to complete the statements about the diagram. the measure of (widehat{ab}) is (\boldsymbol{downarrow}^{circ}). the measure of (angle boc) (\boldsymbol{downarrow}^{circ}). the measure of (widehat{adc}) is (\boldsymbol{downarrow}^{circ}). (the diagram shows a circle with center o. points a, b, c, d are on the circle. (angle aob = 110^{circ}), (angle boc = 36^{circ}). the drop - down menu has options 18, 36, 55, 110.)

Explanation:

Step1: Measure of arc AB

The central angle for arc \( \widehat{AB} \) is given as \( 110^\circ \). The measure of an arc is equal to the measure of its central angle. So, \( m\widehat{AB} = 110^\circ \).

Step2: Measure of \( \angle BCD \)

The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Arc \( \widehat{BD} \) can be found, but first, we know the measure of arc \( \widehat{BC} \) is \( 36^\circ \) (from the diagram, the central angle for \( \widehat{BC} \) is \( 36^\circ \)). Wait, actually, for \( \angle BCD \), it intercepts arc \( \widehat{BD} \)? Wait, no, let's re-examine. Wait, the inscribed angle \( \angle BCD \) intercepts arc \( \widehat{BD} \)? Wait, no, maybe I made a mistake. Wait, the central angle for \( \widehat{AB} \) is \( 110^\circ \), \( \widehat{BC} \) is \( 36^\circ \), so the remaining arc \( \widehat{AD} \) and \( \widehat{DC} \)? Wait, no, the total around a circle is \( 360^\circ \), but maybe we are dealing with a cyclic quadrilateral? Wait, no, let's focus on the first part. Wait, the measure of arc \( \widehat{AB} \) is equal to its central angle, which is \( 110^\circ \), so that's the first answer.

For \( \angle BCD \), it's an inscribed angle intercepting arc \( \widehat{BD} \)? Wait, no, maybe arc \( \widehat{BC} \) is \( 36^\circ \), and arc \( \widehat{AB} \) is \( 110^\circ \), so the arc \( \widehat{AC} \) would be \( 110 + 36 = 146^\circ \), but that's not helpful. Wait, maybe the inscribed angle \( \angle BCD \) intercepts arc \( \widehat{BD} \), but actually, the measure of \( \angle BCD \) is half the measure of arc \( \widehat{BD} \). Wait, but maybe the arc \( \widehat{AB} \) is \( 110^\circ \), so the inscribed angle over arc \( \widehat{AB} \) would be \( 55^\circ \), but wait, the options include 55. Wait, maybe \( \angle BCD \) is an inscribed angle intercepting arc \( \widehat{BD} \), but no, let's check the options. The options are 18, 36, 55, 110. Wait, the measure of \( \angle BCD \): if arc \( \widehat{AB} \) is \( 110^\circ \), then the inscribed angle subtended by arc \( \widehat{AB} \) would be \( \frac{110}{2} = 55^\circ \), so \( \angle BCD = 55^\circ \)? Wait, maybe.

For the measure of arc \( \widehat{ADC} \): the total circumference is \( 360^\circ \), so arc \( \widehat{ADC} \) is the major arc, so \( 360 - \) arc \( \widehat{AC} \)? Wait, no, arc \( \widehat{ADC} \) is the arc from A to D to C, so it's arc \( \widehat{AD} + \widehat{DC} \). But arc \( \widehat{AB} \) is \( 110^\circ \), \( \widehat{BC} \) is \( 36^\circ \), so the remaining arc \( \widehat{AD} + \widehat{DC} = 360 - 110 - 36 = 214^\circ \)? No, that can't be. Wait, maybe I misread the diagram. Wait, the central angle for \( \widehat{AB} \) is \( 110^\circ \), \( \widehat{BC} \) is \( 36^\circ \), so the arc \( \widehat{AC} \) is \( 110 + 36 = 146^\circ \), so the arc \( \widehat{ADC} \) is \( 360 - 146 = 214^\circ \)? No, that's not one of the options. Wait, maybe the diagram is a circle with center O, and points A, B, C, D on the circle. So \( \angle AOB = 110^\circ \), \( \angle BOC = 36^\circ \), so arc \( \widehat{AB} = 110^\circ \), arc \( \widehat{BC} = 36^\circ \), then arc \( \widehat{AD} + \widehat{DC} = 360 - 110 - 36 = 214^\circ \), but that's not an option. Wait, maybe the question is about a different arc. Wait, maybe arc \( \widehat{ADC} \) is the arc from A to D to C, which is the major arc, but maybe I made a mistake. Wait, the options for the third part are 18, 36, 55, 110? No, the dropdown has 18, 36, 55, 110. Wait, maybe the first part: measure of \( \widehat{AB} \) is \( 110^\circ \) (since the central angle is \( 110^\circ \)). Second part: \( \angle BCD \) is an inscribed angle intercepting arc \( \widehat{AB} \), so \( \frac{110}{2} = 55^\circ \). Third part: measure of \( \widehat{ADC} \) – wait, maybe arc \( \widehat{ADC} \) is the sum of arc \( \widehat{AD} \) and \( \widehat{DC} \), but if arc \( \widehat{AB} \) is \( 110^\circ \), arc \( \widehat{BC} \) is \( 36^\circ \), then arc \( \widehat{AD} + \widehat{DC} = 360 - 110 - 36 = 214^\circ \), which is not an option. Wait, maybe the diagram is different. Wait, maybe the circle has center O, and \( \angle AOB = 110^\circ \), \( \angle BOC = 36^\circ \), so arc \( \widehat{AB} = 110^\circ \), arc \( \widehat{BC} = 36^\circ \), then arc \( \widehat{AD} \) and \( \widehat{DC} \) – maybe the inscribed angle \( \angle ADC \) is related, but no. Wait, maybe the first answer is 110 for \( \widehat{AB} \), 55 for \( \angle BCD \) (since 110/2=55), and then the measure of \( \widehat{ADC} \) – wait, maybe \( \widehat{ADC} \) is the arc from A to D to C, which is the major arc, but maybe the question has a typo, or I'm misinterpreting. Wait, let's proceed with the first two:

1. Measure of \( \widehat{AB} \): central angle is \( 110^\circ \), so \( 110^\circ \).

2. Measure of \( \angle BCD \): inscribed angle, half of arc \( \widehat{AB} \) (since \( \angle BCD \) intercepts arc \( \widehat{AB} \)? Wait, no, maybe arc \( \widehat{BD} \) is \( 110^\circ \), so \( \angle BCD = 55^\circ \) (110/2=55).

3. Measure of \( \widehat{ADC} \): maybe the sum of arc \( \widehat{AD} \) and \( \widehat{DC} \), but if arc \( \widehat{AB} \) is \( 110^\circ \), arc \( \widehat{BC} \) is \( 36^\circ \), then arc \( \widehat{AD} + \widehat{DC} = 360 - 110 - 36 = 214^\circ \), which is not an option. Wait, maybe the question is about a different arc, like \( \widehat{ADC} \) is the minor arc? No, that doesn't make sense. Wait, maybe the third part is a mistake, but let's go with the first two:

Answer:

The measure of \( \widehat{AB} \) is \( \boldsymbol{110}^\circ \).
The measure of \( \angle BCD \) is \( \boldsymbol{55}^\circ \).
(For the third part, if we assume a mistake, but based on the options, maybe \( \widehat{ADC} \) is \( 214^\circ \), but that's not in the options. Wait, maybe the diagram is different, and the arc \( \widehat{ADC} \) is the sum of arc \( \widehat{AD} \) and \( \widehat{DC} \), but with the given options, maybe it's a different approach. Alternatively, maybe the first answer is 110, second 55, third 214 (but not in options). Wait, maybe I misread the diagram. Wait, the central angle for \( \widehat{AB} \) is \( 110^\circ \), so arc \( \widehat{AB} = 110^\circ \). The inscribed angle \( \angle BCD \) intercepts arc \( \widehat{AB} \), so \( 110/2 = 55^\circ \). Then the measure of \( \widehat{ADC} \) – maybe it's the arc from A to D to C, which is the major arc, but if the minor arc \( \widehat{AC} \) is \( 110 + 36 = 146^\circ \), then the major arc \( \widehat{ADC} = 360 - 146 = 214^\circ \), but that's not in the options. Wait, maybe the question has a typo, or the diagram is different. Alternatively, maybe the third part is the measure of arc \( \widehat{ADC} \) as \( 214^\circ \), but since it's not in the options, maybe I made a mistake. Alternatively, maybe the arc \( \widehat{ADC} \) is the sum of arc \( \widehat{AD} \) and \( \widehat{DC} \), with arc \( \widehat{AD} \) being \( 110^\circ \) and \( \widehat{DC} \) being \( 104^\circ \), but that's not helpful. Given the options, the first answer is 110, second 55, and maybe the third is 214, but since it's not in the options, maybe the question is different. But based on the given dropdown options (18, 36, 55, 110), the first is 110, second is 55, and maybe the third is 214, but that's not there. Wait, maybe the third part is a mistake, but let's proceed with the first two.