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Find the value of a.

A. 65
B. 130
C. 86
D. 58

To find the value of angle \( a \) in the given circle, we will use properties of circles and isosceles triangles. Let's analyze the given diagram step-by-step:

1. **Identify key angles and relationships:**
   - \( \angle b = 86^\circ \) (given in the diagram)
   - \( \angle d = 29^\circ \) (given in the diagram)
   - \( \angle b \) and \( \angle d \) are subtended by arcs in the circle.
   
2. **Use the fact that angles subtended by the same arc are equal:**
   - Since \( \angle b \) is \( 86^\circ \), and it subtends an arc, any other angle subtended by the same arc (opposite to \( b \) on the circle) will also be \( 86^\circ \). This means angle \( c \), opposite to \( b \), is \( 86^\circ \).

3. **Use the sum of angles in a triangle:**
   - In triangle \( bcd \), the sum of the angles is \( 180^\circ \). Given that \( \angle b = 86^\circ \) and \( \angle d = 29^\circ \), the third angle \( \angle c \) can be found as:
   \[
   \angle c = 180^\circ - \angle b - \angle d = 180^\circ - 86^\circ - 29^\circ = 65^\circ
   \]

4. **Analyze angle relationships around the circle:**
   - Angle \( a \) is opposite to \( \angle c \). Since angles \( a \) and \( c \) subtend the same arc, \( \angle a = \angle c = 65^\circ \).

Thus, the value of \( a \) is \( 65^\circ \).

**Correct Answer:**
\[ \boxed{65} \]

Do you want more details or have any questions?

Here are some related questions you can ask:
1. How do you find the angle subtended by an arc in a circle?
2. What are the properties of isosceles triangles?
3. How can you use the exterior angle theorem in a circle?
4. What are cyclic quadrilaterals, and how do their angles relate?
5. How do the properties of supplementary angles apply in circles?

**Tip:** Remember, in a circle, angles subtended by the same arc are always equal.

Learn how to find the value of angle a in a circle using properties of circle geometry and isosceles triangles. Detailed step-by-step solution provided with angles b = 86° and d = 29°. Suitable for high school students studying circle geometry.

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