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Find the value of b.

A. 50
B. 61.5
C. 123
D. 57

To find the value of \( b \), we need to use the properties of circles and the angles related to them.

Given:
- The angle at the external tangent point is \( 57^\circ \).
- The angle inside the quadrilateral opposite the external angle is \( 100^\circ \).

1. **Identify the related angles:**
   - The external angle \( 57^\circ \) and the internal angle \( b \) at the tangent point are supplementary because they form a straight line.
   - This means \( b = 180^\circ - 57^\circ \).

2. **Calculate \( b \):**
   \[
   b = 180^\circ - 57^\circ = 123^\circ
   \]

So, the correct answer is \( b = 123^\circ \) which corresponds to option C.

Would you like more details or have any questions about this solution? Here are some questions you might ask next:

1. How can we determine the angle \( a \) in the diagram?
2. What are the properties of angles in a cyclic quadrilateral?
3. How do we calculate angles using tangent properties in a circle?
4. Can you explain the significance of supplementary angles in this context?
5. What are some other key properties of circles related to tangents and angles?

**Tip:** Remember, the sum of angles around a point is always \( 360^\circ \), which can help solve many geometric problems.

Learn how to find the value of angle b in a circle problem using circle properties and supplementary angles. This problem involves geometric concepts related to circles and angles. Suitable for high school students.

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