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### 1. **Calculate the missing angles:**

#### (a)
We have a cyclic quadrilateral, which means opposite angles in the quadrilateral add up to 180°.

- Given: 
  - One angle is 82° (angle at the bottom left).
  - One angle is 116° (angle at the bottom right).
  
We need to find angles \( a \) and \( b \).

- **Finding angle \( a \)**: 
  Since opposite angles in a cyclic quadrilateral add up to 180°,
  \[
  a + 116^\circ = 180^\circ
  \]
  \[
  a = 180^\circ - 116^\circ = 64^\circ
  \]

- **Finding angle \( b \)**:
  Similarly, for \( b \),
  \[
  b + 82^\circ = 180^\circ
  \]
  \[
  b = 180^\circ - 82^\circ = 98^\circ
  \]

#### (b)
This diagram involves tangents from an external point to the circle, and angles between tangents and radii drawn to the point of contact. We are given:
- One angle is 52° (angle between one tangent and the line from \( X \)).
- The tangent angles are equal because the tangents drawn from a point outside the circle are equal in length.

- **Finding angle \( c \)**:
  The angle between a tangent and a radius is always 90°. Therefore,
  \[
  c = 90^\circ
  \]

- **Finding angle \( d \)**:
  Using the fact that angle \( d \) is supplementary to the 52° angle (since they form a straight line),
  \[
  d + 52^\circ = 180^\circ
  \]
  \[
  d = 180^\circ - 52^\circ = 128^\circ
  \]

### Final Answers:
- \( a = 64^\circ \)
- \( b = 98^\circ \)
- \( c = 90^\circ \)
- \( d = 128^\circ \)

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Would you like more details or have any questions on this solution? Here are some related questions to expand your understanding:

1. What is the property of opposite angles in a cyclic quadrilateral?
2. Why is the angle between a tangent and a radius always 90°?
3. How can you prove that tangents from the same external point are equal in length?
4. What other methods can you use to find missing angles in a circle involving tangents?
5. How do you determine if a quadrilateral is cyclic?

**Tip:** In cyclic quadrilaterals, opposite angles always add up to 180°.

The original question involves finding missing angles in a cyclic quadrilateral and angles formed by tangents and radii.

Learn how to solve a geometry problem involving a cyclic quadrilateral and tangents to a circle. This guide explains how to find missing angles using the properties of cyclic quadrilaterals and tangents. Suitable for students in Grades 9-10, this step-by-step explanation covers key theorems such as the Cyclic Quadrilateral Theorem and Tangent-Radius Theorem.

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