Let's solve the questions from the image you provided.

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:

• $a = 64^\circ$
• $b = 98^\circ$
• $c = 90^\circ$
• $d = 128^\circ$

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°.