Question 9: Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD.

**Answer:**Let us join A to P to make chord AP and then join D to Q to make chord DQ

For chord AP;

∠ACP = ∠PBA ………….(1)

(Angles in the same segment)

Similarly, for chord DQ;

∠DCQ = ∠DBQ …………(2)

We also have;

∠PBA = ∠DBQ (Vertically opposite angles) …………..(3)

From equations (1), (2) and (3);

∠ACP = ∠DCQ proved

Question 10: If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

**Answer:** ABC is a triangle. AB and AC are diameters of two circles which intersect at A and D.

∠ADC = ∠ADB = 90° (angles in semi-circle)

As ∠ADC + ∠ADB = 180°So, AB is a straight line

This proves that D lies on AB and is the point of intersection for two circles.

Question 11: ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

**Answer:** Given ABC and ADC are right triangles with common hypotenuse AC.

To Prove ∠CAD = ∠CBD

Let us take chord DC;

∠CAD = ∠CBD

(Angles in the same segment).

Question 12: Prove that a cyclic parallelogram is a rectangle.

**Answer:** Opposite angles of a cyclic parallelogram are supplementary.

If opposite angles are ∠1 and ∠2

Then; ∠1 + ∠2 = 180°

Moreover, opposite angles of a cyclic parallelogram are equal

Hence, ∠1 = ∠2 = 90°

Thus all angles of cyclic parallelogram are right angles

Hence, a cyclic parallelogram is a rectangle proved.

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