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Find the measure of b.
A. 20
B. 80
C. 40
D. 70

To find the measure of angle \( b \) in the given figure, let's analyze the relationships and properties of the angles in the circle.

1. We know the angle \( 20^\circ \) at one of the vertices of the triangle.
2. The angles \( a \), \( b \), \( c \), \( d \), and \( e \) are within the circle, forming various geometric properties with the center \( O \).

Given that the sum of angles in a triangle is \( 180^\circ \), and the central angles are related to the inscribed angles, we can find the measure of angle \( b \) using circle theorems.

First, we note that angle \( c \) is an external angle to the triangle formed by \( 20^\circ \) and \( b \) at \( O \):
\[ c = 180^\circ - (20^\circ + b) \]

Second, angle \( c \) is subtended by the arc opposite to \( d \) and \( e \). We know that the sum of angles around point \( O \) is \( 360^\circ \):
\[ c + b + e = 180^\circ \]

Since angles at the center are twice the inscribed angles on the circle, considering opposite arcs, we conclude for angle:
\[ b = 20^\circ \]

Because the angles within circle \[ b= \]

Hence, logically fitting value:
\[ \boxed{80^\circ} \]

Since circle symmetry leads angles consideration consistent, checked choices aligns thusly:

Would you like further detail on this problem or have any questions?

Here are five questions you might ask:
1. How do inscribed angles relate to central angles in a circle?
2. What is the relationship between external and internal angles in a circle?
3. How can the properties of triangles help solve circle problems?
4. What are some key circle theorems used in geometry?
5. How do you use the sum of angles in various geometric shapes to solve problems?

**Tip:** When solving problems involving circles, always check the relationships between central and inscribed angles, and remember that the sum of angles around a point is \(360^\circ\).

Learn how to find the measure of angle b in a circle using circle theorems. This problem involves analyzing central and inscribed angles, applying the sum of angles in a triangle, and understanding geometric properties of circles. Suitable for high school geometry students.

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