Question

in circle b with the measure of minor arc \\(\overarc{ac} = 126^\circ\\), find \\(m\angle adc\\).

Explanation:

Step1: Find the measure of major arc \( \overarc{AC} \)

The total measure of a circle is \( 360^\circ \). The minor arc \( \overarc{AC} = 126^\circ \), so the major arc \( \overarc{AC} = 360^\circ - 126^\circ = 234^\circ \).

Step2: Use the inscribed angle theorem for a cyclic quadrilateral (or the property of an inscribed angle subtended by an arc)

Wait, actually, \( \angle ADC \) is an inscribed angle subtended by the major arc \( \overarc{AC} \)? No, wait. Wait, the angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, no, in this case, \( DC \) and \( DA \): \( \angle ADC \) is an angle formed by two chords, but actually, the measure of an inscribed angle is half the measure of its intercepted arc. Wait, no, if we consider the arc that is opposite to the angle. Wait, the key here is that the measure of an angle formed by a chord and a tangent is half the measure of the intercepted arc, but here \( DC \) is a chord, \( DA \) is a chord. Wait, maybe I made a mistake. Wait, the minor arc \( AC \) is \( 126^\circ \), so the arc \( ADC \) (the major arc) is \( 360 - 126 = 234^\circ \). But \( \angle ADC \) is an inscribed angle? Wait, no, point \( D \) is on the circle, \( A \) and \( C \) are on the circle. So \( \angle ADC \) is an inscribed angle that intercepts arc \( AC \). Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. But if the angle is on the circumference, and the arc is the one opposite. Wait, actually, when the angle is on the circumference, the intercepted arc is the one that is not containing the angle. Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. So if the angle is \( \angle ADC \), the intercepted arc is \( \overarc{AC} \). But wait, if the angle is on the circumference, and the arc is minor or major. Wait, no, the measure of the inscribed angle is half the measure of the arc it intercepts. But if the angle is in the major segment, the intercepted arc is the minor arc, and if it's in the minor segment, the intercepted arc is the major arc. Wait, in this case, point \( D \) is on the circle, and the arc \( AC \) is minor (\( 126^\circ \)), so the angle \( \angle ADC \) is an inscribed angle that intercepts the major arc \( AC \)? No, wait, no. Wait, let's recall: the measure of an angle formed by two chords in a circle is equal to half the sum of the measures of the intercepted arcs. Wait, no, that's for angles inside the circle. For angles on the circumference (inscribed angles), the measure is half the measure of the intercepted arc. Wait, I think I messed up. Let's start over.

The total circumference is \( 360^\circ \). The minor arc \( AC \) is \( 126^\circ \), so the major arc \( AC \) is \( 360 - 126 = 234^\circ \). Now, \( \angle ADC \) is an inscribed angle that intercepts the major arc \( AC \)? No, wait, no. Wait, the angle at \( D \), with chords \( DC \) and \( DA \). The intercepted arc for \( \angle ADC \) is arc \( AC \). But if the angle is on the circumference, the measure of the angle is half the measure of the intercepted arc. But if the angle is in the major segment, the intercepted arc is the minor arc, and the angle is acute. Wait, no, let's take an example. If the arc \( AC \) is \( 126^\circ \), then an inscribed angle intercepting arc \( AC \) would be \( \frac{126}{2} = 63^\circ \), but that's if the angle is on the opposite side. Wait, no, actually, the measure of an inscribed angle is half the measure of its intercepted arc. So if the angle is on the circumference, and the arc is the one that is not between the two sides of the angle. Wait, maybe the correct approach is: the angle \( \angle ADC \) is formed by two chords \( DC \) and \( DA \), and the intercepted arc is \( AC \). But since \( D \) is on the circle, and the arc \( AC \) is minor (\( 126^\circ \)), the angle \( \angle ADC \) is actually an angle that is subtended by the major arc \( AC \). Wait, no, I think I made a mistake. Let's recall the property: the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc, but here \( DC \) is a chord, not a tangent. Wait, looking at the diagram, \( DC \) is a short chord, and \( DA \) is a longer chord. Wait, maybe the angle \( \angle ADC \) is an angle formed by two chords, and the measure of the angle is half the difference of the measures of the intercepted arcs. Wait, no, that's for angles inside the circle. Wait, no, the formula for an angle formed by two chords intersecting at a point on the circle (an inscribed angle) is half the measure of the intercepted arc. Wait, I think I was confused earlier. Let's use the correct theorem: the measure of an inscribed angle is half the measure of its intercepted arc. So if \( \angle ADC \) is an inscribed angle intercepting arc \( AC \), then \( m\angle ADC = \frac{1}{2} \times \) measure of arc \( AC \). But wait, arc \( AC \) is \( 126^\circ \), so \( \frac{126}{2} = 63^\circ \)? But that doesn't seem right. Wait, no, maybe the angle is intercepting the major arc. Wait, no, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. So if the angle is on the circumference, and the arc is the one that is opposite to the angle. So if the arc \( AC \) is \( 126^\circ \), then the angle \( \angle ADC \) (which is on the circumference) intercepts arc \( AC \), so its measure is half of \( 126^\circ \)? But that would be \( 63^\circ \), but that seems too small. Wait, no, maybe I have the arc wrong. Wait, the total circle is \( 360^\circ \), so the arc from \( A \) to \( C \) through \( D \) is \( 360 - 126 = 234^\circ \). Then, the angle \( \angle ADC \) intercepts the arc \( AC \) that is not containing \( D \). Wait, no, the intercepted arc is the arc that is between the two sides of the angle. So \( \angle ADC \) has sides \( DC \) and \( DA \), so the intercepted arc is \( AC \). So the measure of \( \angle ADC \) is half the measure of arc \( AC \). But arc \( AC \) is \( 126^\circ \), so \( \frac{126}{2} = 63^\circ \). Wait, but that seems incorrect. Wait, maybe the angle is a reflex angle? No, angles in a circle are typically less than \( 180^\circ \). Wait, no, I think I made a mistake. Let's check the theorem again. The inscribed angle theorem states that an angle \( \theta \) subtended by an arc \( s \) at the circumference of a circle is half the central angle subtended by the same arc. So if the central angle for arc \( AC \) is \( 126^\circ \) (since it's a minor arc), then the inscribed angle subtended by arc \( AC \) would be \( \frac{126}{2} = 63^\circ \). But in this case, the angle \( \angle ADC \) is subtended by arc \( AC \), so it should be \( 63^\circ \). Wait, but that seems too small. Wait, maybe the diagram is different. Wait, the diagram shows point \( D \) below \( C \), and \( A \) above. So \( DC \) is a short chord, \( DA \) is a long chord. So \( \angle ADC \) is an angle at \( D \), between \( DC \) and \( DA \). So the arc \( AC \) is the minor arc, and the angle \( \angle ADC \) is an inscribed angle intercepting the minor arc \( AC \), so its measure is half of \( 126^\circ \), which is \( 63^\circ \). Wait, but that seems correct. Wait, no, wait a second. The measure of an angle formed by two chords in a circle is equal to half the sum of the measures of the intercepted arcs. Wait, no, that's when the angle is inside the circle. If the angle is on the circumference, it's half the measure of the intercepted arc. If the angle is outside the circle, it's half the difference of the intercepted arcs. So in this case, \( \angle ADC \) is on the circumference, so it's half the measure of the intercepted arc \( AC \). So \( m\angle ADC = \frac{1}{2} \times 126^\circ = 63^\circ \)? Wait, no, that can't be. Wait, no, the intercepted arc for \( \angle ADC \) is actually the arc \( AC \) that is opposite to the angle. Wait, maybe I got the arc wrong. Let's calculate the major arc \( AC \): \( 360 - 126 = 234^\circ \). Then, the angle \( \angle ADC \) is an inscribed angle intercepting the major arc \( AC \), so its measure is \( \frac{1}{2} \times 234^\circ = 117^\circ \). Ah! That makes sense. Because if the angle is on the circumference, and the arc is the major arc, then the angle is obtuse. So the correct approach is: the measure of an inscribed angle is half the measure of its intercepted arc. If the angle is in the major segment (i.e., the segment that contains the major arc), then the intercepted arc is the minor arc, and the angle is acute. If the angle is in the minor segment, the intercepted arc is the major arc, and the angle is obtuse. In this case, point \( D \) is in the minor segment (since the minor arc \( AC \) is \( 126^\circ \), the minor segment is the smaller area), so the angle \( \angle ADC \) intercepts the major arc \( AC \), so its measure is half of \( 234^\circ \), which is \( 117^\circ \).

So let's redo the steps:

Step1: Calculate the measure of the major arc \( \overarc{AC} \)

The total measure of a circle is \( 360^\circ \). The minor arc \( \overarc{AC} = 126^\circ \), so the major arc \( \overarc{AC} = 360^\circ - 126^\circ = 234^\circ \).

Step2: Use the inscribed angle theorem to find \( m\angle ADC \)

The measure of an inscribed angle is half the measure of its intercepted arc. Here, \( \angle ADC \) is an inscribed angle that intercepts the major arc \( \overarc{AC} \). So:
\[ m\angle ADC = \frac{1}{2} \times \text{measure of major arc } \overarc{AC} = \frac{1}{2} \times 234^\circ = 117^\circ \]

Answer:

\( 117^\circ \)