student-name Diya Singh asked in MathProve that equal chords of a circle subtend equal angles at the centre. 1 Follow 0student-name Aakash Sharma answered this4624 helpful votes in Math, Class XI-ScienceHere is the link for the answer to your query. http://www.meritnation.com/ask-answer/question/prove-that-equal-chords-of-a-circle-subtend-equal-angles-at/circles/1530288Was this answer helpful4100% users found this answer helpful.student-name Ishan Goyal answered this1083 helpful votes in Math, Class XI-CommerceConsider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.In ΔAOB and ΔCO'D,AB = CD (Chords of same length)OA = O'C (Radii of congruent circles)OB = O'D (Radii of congruent circles)∴ ΔAOB ≅ ΔCO'D (SSS congruence rule)⇒ ∠AOB = ∠CO'D (By CPCT)Hence, equal chords of congruent circles subtend equal angles at their centres

Proof of Equal Angles at the Centre Subtending Equal Chords of a Circle:

1. Definition of Equal Angles:
- Two angles are said to be equal if they have the same measure.

2. Equal Angles at the Centre:
- When two chords of a circle subtend equal angles at the center, then those chords are equal in length.

3. Proof:
- Let us consider a circle with center O and two chords AB and CD that subtend equal angles ∠AOB and ∠COD at the center.
- Draw radii OA and OC to the points A and C respectively.
- Since OA = OC (radii of the same circle), triangle OAB is congruent to triangle OCD by Side-Angle-Side (SAS) congruence.
- Therefore, angle AOB = angle COD (corresponding parts of congruent triangles).
- Now, in a circle, an angle at the center is double the angle at the circumference subtended by the same arc.
- So, angle AOB = 2*arc AB and angle COD = 2*arc CD.
- Since angle AOB = angle COD, we have 2*arc AB = 2*arc CD.
- Dividing both sides by 2, we get arc AB = arc CD.
- Hence, the chords AB and CD are equal in length.

Therefore, when two chords of a circle subtend equal angles at the center, they are equal in length.