1. Two circles touch each other externally at point C. AB is a common tangent to both circles, touching them at points A and B respectively. Find the measure of ∠ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

2. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

3. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

4. Two circles touch externally at point C. AB is a common tangent to both circles, touching them at points A and B. The measure of \(\angle\) ACB is?

(a) 60° (b) 45° (c) 90° (d) 75°

5. Two circles are externally tangent at point \( C \). \( AB \) is a common tangent to both circles, touching them at points \( A \) and \( B \). Determine the measure of \( \angle ACB \).

(a) 60° (b) 90° (c) 45° (d) 30°

6. Two circles touch externally at point \(C\). A common external tangent to both circles touches them at points \(A\) and \(B\) respectively. The measure of \(\angle ACB\) is -

(a) 60° (b) 45° (c) 90° (d) 30°

7. Two circles centered at P and Q externally touch each other at point A. A common external tangent to both circles touches them at points R and S, respectively. Prove that \(\angle\)RAS = 90°.

8. Two circles touch each other externally at point C. A common tangent AB touches the circles at points A and B respectively. Find the value of \(\angle ACB\).

9. If two circles touch each other externally at point A, and the number of direct common tangents is \(= x\), and the number of transverse common tangents is \(= y\), then \(x - y =-----\).

10. Two circles touch each other externally at point \( C \). A common external tangent touches the two circles at points \( A \) and \( B \). Determine the value of \( \angle ACB \).

(a) 60° (b) 45° (c) 30° (d) 90°

11. I have drawn two circles with centers A and B that externally touch each other at point O. A straight line is drawn through point O, which intersects the two circles at points P and Q respectively. Prove that AP is parallel to BQ.

12. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.