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To find the measure of angle \( a \) in the given circle, we can use the properties of cyclic quadrilaterals and the fact that the sum of the interior angles of any triangle is \( 180^\circ \).

In the diagram:
- Triangle \( bc \) is isosceles because the two sides are marked with equal lengths, making the base angles equal.
- \( \angle bdc = 86^\circ \) and since it's the angle opposite the base, the base angles are \( \frac{180^\circ - 86^\circ}{2} = 47^\circ \).

We have:
- \( \triangle OaB \) is isosceles (as indicated by the marks), and we need to find \( \angle a \).
- The angle \( \angle OaB \) that subtends the arc \( bd \) can be calculated as follows: Since \( \triangle OaB \) is isosceles and \( \angle OaB = 2 \times 29^\circ \), we have the following:

\[
\angle OaB = 2 \times 29^\circ = 58^\circ
\]

\( \angle a \) is the angle opposite to \( \angle OaB \), so:
\[
\angle a + \angle OaB = 180^\circ
\]
\[
\angle a + 58^\circ = 180^\circ
\]
\[
\angle a = 180^\circ - 58^\circ
\]
\[
\angle a = 122^\circ
\]

So, the value of \( \angle a \) is \( 122^\circ \).

Would you like more details or have any questions?

Here are some additional questions you might find interesting:
1. What is the significance of the angle subtended by the diameter of a circle?
2. How do you determine the angle measures in an isosceles triangle?
3. What is the relationship between angles and arcs in a circle?
4. How can the properties of cyclic quadrilaterals be applied in circle geometry?
5. How do you find the measure of an angle formed by intersecting chords in a circle?

**Tip:** In circle geometry, knowing the properties of special quadrilaterals like cyclic quadrilaterals (quadrilaterals inscribed in a circle) can greatly simplify finding angle measures.

Learn how to find the measure of angle 'a' in circle geometry using properties of isosceles triangles, cyclic quadrilaterals, and the sum of interior angles of a triangle theorem. Detailed step-by-step solution provided.

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