What is the value of the angle marked with '?' in the diagram?

This is a geometry problem involving angles in a circle and triangles.

Let's break it down:

1. Identify known angles and arcs:
   • We are given that angle B is 42 degrees. This angle is formed by a tangent (line segment from B to A) and a secant (line segment from B to E passing through D).
   • The arc CE is given as 110 degrees.
2. Relationship between inscribed angle and intercepted arc: An inscribed angle is half the measure of its intercepted arc. However, angle B is not an inscribed angle.
3. Angle formed by a tangent and a secant: The measure of an angle formed by a tangent and a secant drawn from an external point to a circle is half the difference of the measures of the intercepted arcs. In this case, angle B intercepts arc AC and arc DE. So, \( \angle B = \frac{1}{2} (m\stackrel{\frown}{AC} - m\stackrel{\frown}{DE}) \).
4. Angle formed by two secants: If we consider the point where the secant from B intersects the circle at A and E, and the other secant from D to A and some other point on the circle, this formula becomes more complex. Let's reconsider the given diagram. It appears that B is an external point from which a tangent AB and a secant BDE are drawn. However, the line segment from B to A is shown as a secant passing through the circle. Let's assume that AB is a chord and B is an external point from which a secant is drawn. The diagram is a bit ambiguous.
5. Re-interpreting the diagram: Let's assume A, D, E are points on the circle, and O is the center. B is an external point. The line segment BA is a secant intersecting the circle at A and another point (not labeled). The line segment BDE is a secant intersecting the circle at D and E. Angle B = 42 degrees. The arc CE = 110 degrees. We need to find the angle marked with '?'. This angle is likely referring to an angle at point D or some other point within the circle.
6. Alternative interpretation: Let's assume that BA is a tangent at A, and BDE is a secant. Angle B = 42 degrees. Arc CE = 110 degrees. We need to find the angle marked '?'. The '?' is near point D, likely referring to angle ADE or angle ADB or angle BDA or an arc. Given the lines drawn, it is most likely asking for an angle related to triangle ABD or triangle BDE or within the circle.
7. Let's assume the question is asking for the measure of arc AD. If BA is a secant and BDE is a secant, then angle B = 42 degrees. We have arc CE = 110 degrees.
8. Let's assume AB is a chord, and the line segment from B passes through D and E, and also through A. This would make BDE and BAE secants. This is unlikely.
9. Let's assume the diagram shows: A, D, E, C are points on the circle. B is an external point. BA is a secant. BDE is a secant. Angle B = 42 degrees. Arc CE = 110 degrees. The '?' is near point D and it seems to be asking for the measure of arc AD.
10. If BA is a secant and BDE is a secant, then \( \angle B = \frac{1}{2} |m\stackrel{\frown}{AE} - m\stackrel{\frown}{CD}| \) or \( \angle B = \frac{1}{2} |m\stackrel{\frown}{AE} - m\stackrel{\frown}{DE}| \) or \( \angle B = \frac{1}{2} |m\stackrel{\frown}{AC} - m\stackrel{\frown}{DE}| \). The diagram is not clear about which arcs are intercepted.
11. Let's assume A, D, E, C are points on the circle, O is the center. Line segment BAE is a secant and BDC is a secant. Angle B = 42 degrees. Arc CE = 110 degrees. The '?' is likely referring to arc AD. If this is the case, then \( \angle B = \frac{1}{2} (m\stackrel{\frown}{CE} - m\stackrel{\frown}{AD}) \). So, \( 42^{\circ} = \frac{1}{2} (110^{\circ} - m\stackrel{\frown}{AD}) \). This implies \( 84^{\circ} = 110^{\circ} - m\stackrel{\frown}{AD} \). Then \( m\stackrel{\frown}{AD} = 110^{\circ} - 84^{\circ} = 26^{\circ} \).
12. Let's consider another common configuration: A and E are points on the circle. B is an external point. BA is a tangent at A. BDE is a secant. Angle B = 42 degrees. Arc CE = 110 degrees. The '?' is at D and likely asking for arc AD. In this case, the angle formed by a tangent and a secant is \( \angle B = \frac{1}{2} (m\stackrel{\frown}{AE} - m\stackrel{\frown}{DE}) \). This is not helpful without knowing arc AE and DE.
13. Let's assume AB is a chord, and BDE is a secant. Angle at B is 42 degrees. Arc CE = 110 degrees. The '?' is asking for arc AD. This setup is not standard.
14. Let's go back to the interpretation where BAE and BDC are secants. So, B, A, E are collinear, and B, D, C are collinear. This also seems incorrect from the diagram.
15. Most probable interpretation: A, D, E, C are points on the circle. B is an external point. BA is a secant. BDE is a secant. Angle B = 42 degrees. Arc CE = 110 degrees. The '?' is asking for arc AD. In this scenario, the intercepted arcs for angle B are arc CE and arc AD. Therefore, \( \angle B = \frac{1}{2} (m\stackrel{\frown}{CE} - m\stackrel{\frown}{AD}) \).
16. Plugging in the values: \( 42^{\circ} = \frac{1}{2} (110^{\circ} - m\stackrel{\frown}{AD}) \).
17. Multiply both sides by 2: \( 84^{\circ} = 110^{\circ} - m\stackrel{\frown}{AD} \).
18. Solve for m\(\stackrel{\frown}{AD}\): \( m\stackrel{\frown}{AD} = 110^{\circ} - 84^{\circ} \).
19. Calculate the result: \( m\stackrel{\frown}{AD} = 26^{\circ} \).

Therefore, assuming the question asks for the measure of arc AD and BAE and BDC are secants:

Ответ: 26°