text: # Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
  
  Given:
  
  Chords of congruent circles subtend equal angles at their centres.
  
  To do:
  
  We have to prove that the chords are equal.
  
  Solution:
  
  Let $c_{1}$ and $C_{2}$ be two congruent circles, $AB$ and $PQ$ are their chords respectively.
  
  Let us join $OA$ and $OB$ in circle $C_{1}$.
  
  Similarly, in cirlcle $C_{2}$, join $MP$ and $MQ$.
  
  In $\vartriangle OAB$ and $\vartriangle MPQ$.
  
  $OA=MP$                                [$\because$ Radius of the congruent circles are same]
  
  $OB=MQ$                               [$\because$ Radius of congruent circles are same]
  
  $\angle AOB=\angle PMQ$              [It is given chords of congruent circles subtend equal angles at their centres]
  
  $\Rightarrow \vartriangle OAB\cong \vartriangle MPQ$                [SAS rule of congruency]
  
  $\therefore AB=PQ$                           [By CPCT rule]
  
  Hence, it has been proved that the chords are equal.
  
  Tutorialspoint
  
  Simply Easy Learning
  
  Updated on: 10-Oct-2022
  
  27 Views
explanation: The document contains a mathematical proof. It outlines the given conditions and the goal, and then provides a step-by-step solution to prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal. The document does not contain a question but rather a complete proof.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
  
  Given:
  
  Chords of congruent circles subtend equal angles at their centres.
  
  To do:
  
  We have to prove that the chords are equal.
  
  Solution:
  
  Let $c_{1}$ and $C_{2}$ be two congruent circles, $AB$ and $PQ$ are their chords respectively.
  
  Let us join $OA$ and $OB$ in circle $C_{1}$.
  
  Similarly, in cirlcle $C_{2}$, join $MP$ and $MQ$.
  
  In $\vartriangle OAB$ and $\vartriangle MPQ$.
  
  $OA=MP$                                [$\because$ Radius of the congruent circles are same]
  
  $OB=MQ$                               [$\because$ Radius of congruent circles are same]
  
  $\angle AOB=\angle PMQ$              [It is given chords of congruent circles subtend equal angles at their centres]
  
  $\Rightarrow \vartriangle OAB\cong \vartriangle MPQ$                [SAS rule of congruency]
  
  $\therefore AB=PQ$                           [By CPCT rule]
  
  Hence, it has been proved that the chords are equal.
  
  Tutorialspoint
  
  Simply Easy Learning
  
  Updated on: 10-Oct-2022
  
  27 Views
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a mathematical proof. It outlines the given conditions and the goal, and then provides a step-by-step solution to prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal. The document does not contain a question but rather a complete proof.
  
  Label: no-question