The problem provided involves a cyclic quadrilateral $ABCD$ and a circle with a tangent at point $A$. Let's break this down and solve for the angles asked.

Information provided:

• $ABCD$ is a cyclic quadrilateral (which means opposite angles sum up to 180°).
• There is a tangent at point $A$ making an angle of 70° with the side $AD$.
• $\angle BCA = 30^\circ$.

We are tasked with finding the sizes of:

1. $\angle ACD$
2. $\angle BAD$

a) Finding $\angle ACD$:

We are given that $\angle BCA = 30^\circ$. Since $ABCD$ is a cyclic quadrilateral, opposite angles of a cyclic quadrilateral sum to 180°: $\angle BCA + \angle ACD = 180^\circ$ Substitute the known value of $\angle BCA$: $30^\circ + \angle ACD = 180^\circ$ $\angle ACD = 180^\circ - 30^\circ = 150^\circ$

b) Finding $\angle BAD$:

To find $\angle BAD$, we use the fact that the angle between the tangent at point $A$ and the chord $AD$ is equal to the angle subtended by the chord $AD$ at the opposite side of the circle (alternate segment theorem). The angle formed between the tangent and the chord $AD$ is $70^\circ$, which means: $\angle BAD = 70^\circ$

Final Answers:

• $\angle ACD = 150^\circ$
• $\angle BAD = 70^\circ$

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. What is the significance of a cyclic quadrilateral in terms of its angle properties?
2. How does the alternate segment theorem apply to different circle theorems?
3. Can you determine $\angle ABC$ using the information from this problem?
4. How would you find other unknown angles in the quadrilateral if more angles were missing?
5. Why is the sum of opposite angles in a cyclic quadrilateral always 180°?

Tip:

When working with cyclic quadrilaterals, always remember the key property: opposite angles are supplementary (sum to 180°). This is a powerful tool for solving problems involving circles and quadrilaterals.