To solve for angle $a^\circ$ in this diagram, we can analyze the geometry based on the information provided:

1. Identified Angles: There are two angles marked as $30^\circ$ and $70^\circ$.
2. Circle and Central Angle: Since $O$ is the center of the circle, the line segments from $O$ to any point on the circle are radii.
3. Central Angle and Inscribed Angle Relationship: In a circle, the angle subtended at the center by a chord is twice the angle subtended on the circumference by the same chord.

Solution Steps

• The two given angles, $30^\circ$ and $70^\circ$, add up to create a central angle of $100^\circ$.
• The inscribed angle $a$, subtended by this arc, will be half of this central angle.

$a = \frac{100^\circ}{2} = 50^\circ$

Conclusion

The correct answer is: $a = 50^\circ$

Would you like further details or explanations on this solution?

Here are some related questions:

1. How do we identify inscribed and central angles in a circle?
2. What is the relationship between a central angle and an inscribed angle in a circle?
3. How do the properties of isosceles triangles relate to circle geometry?
4. Can this type of angle calculation be applied to other polygons?
5. Why is the sum of angles in a triangle always $180^\circ$?

Tip: Remember, in circles, inscribed angles are always half the measure of their corresponding central angles subtending the same arc.