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# Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres

**Solution:**

Let QR and YZ be the equal chords of 2 congruent circles.

Then, QR = YZ

We need to prove that they subtend equal angles at the center. That is, ∠QPR = ∠YXZ

We know that the radii of both circles are equal. So, we get: PR = PQ = XZ = XY

Consider the 2 triangles, ∆PQR and ∆XYZ.

PQ = XY (Radii are equal)

PR = XZ (Radii are equal)

QR = YZ (Chords are equal)

By SSS congruency, ∆PQR is congruent to ∆XYZ.

So, by CPCT (Corresponding parts of congruent triangles), we get ∠QPR =∠YXZ.

Hence, proved that equal chords of congruent circles subtend equal angles at their centers.

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

**Video Solution:**

## Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres

Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.2 Question 1

**Summary:**

If two circles are congruent, they have the same radii, then equal chords of congruent circles subtend equal angles at their centers

**☛ Related Questions:**

- Fill in the blanks:(i) The center of the circle lies in____of the circle. (exterior / interior)(ii) A point, whose distance from the center of circle is greater than its radius lies in____of the circle. (exterior / interior)(iii) The longest chord of the circle is a_____of the circle.iv) An arc is a______when its ends are the ends of a diameter.(v) Segment of a circle is the region between an arc andof the circle.(vi) A circle divides the plane, on which it lies, inparts.
- Write True or False: Give reasons for your answers.(i) Line segment joining the centre to any point on the circle is a radius of the circle.(ii) A circle has only finite number of equal chords.(iii) If a circle is divided into three equal arcs, each is a major arc.(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.(v) Sector is the region between the chord and its corresponding arc.(vi) A circle is a plane figure.
- Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.

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