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# 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, P and Q respectively (see Fig. 10.40). Prove that ∠ACP = ∠QCD.

**Solution:**

∠ACP and ∠ABP lie in the same segment. Similarly, ∠DCQ and ∠DBQ lie in the same segment.

We know that angles in the same segment of a circle are equal.

So, we get ∠ACP = ∠ABP and ∠QCD = ∠QBD

Also, ∠QBD = ∠ABP (Vertically opposite angles)

Therefore, ∠ACP = ∠QCD.

**☛ Check: **NCERT Solutions for Class 9 Maths Chapter 10

**Video Solution:**

## 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, P and Q respectively (see Fig. 10.40). Prove that ∠ACP = ∠QCD

Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.5 Question 9

**Summary:**

If 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, P, and Q respectively, then ∠ACP = ∠QCD.

**☛ Related Questions:**

- 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.
- ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD
- Prove that a cyclic parallelogram is a rectangle.
- Prove that the line of centers of two intersecting circles subtends equal angles at the two points of intersection.

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