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  Guru
  
  # Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. Q.5
  
  • 0
  
  Hello sir i want to know the best solution of the question from exercise 8.1of math of Quadrilaterals chapter of class 9th give me the best and easy for solving this question how i solve it of question no. 5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
  
  Share
  
  1. Given that,
  
  Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at right angle at O.
  
  To prove that,
  
  The Quadrilateral ABCD is a square.
  
  Proof,
  
  In ΔAOB and ΔCOD,
  
  AO = CO (Diagonals bisect each other)
  
  ∠AOB = ∠COD (Vertically opposite)
  
  OB = OD (Diagonals bisect each other)
  
  , ΔAOB ≅ ΔCOD [SAS congruency]
  
  Thus,
  
  AB = CD [CPCT] — (i)
  
  also,
  
  ∠OAB = ∠OCD (Alternate interior angles)
  
  ⇒ AB || CD
  
  Now,
  
  In ΔAOD and ΔCOD,
  
  AO = CO (Diagonals bisect each other)
  
  ∠AOD = ∠COD (Vertically opposite)
  
  OD = OD (Common)
  
  , ΔAOD ≅ ΔCOD [SAS congruency]
  
  Thus,
  
  AD = CD [CPCT] — (ii)
  
  also,
  
  ⇒ AD = BC = CD = AB — (ii)
  
  and ∠ADC+∠BCD = 180° (co-interior angles)
explanation: The text extract contains a math question and a detailed solution to the question. The question asks to prove that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then the quadrilateral is a square. The solution provided uses geometric properties and congruency theorems to prove the statement.
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.
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distilabel_metadata.role: system
distilabel_metadata.content: • 0
  Guru
  
  # Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. Q.5
  
  • 0
  
  Hello sir i want to know the best solution of the question from exercise 8.1of math of Quadrilaterals chapter of class 9th give me the best and easy for solving this question how i solve it of question no. 5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
  
  Share
  
  1. Given that,
  
  Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at right angle at O.
  
  To prove that,
  
  The Quadrilateral ABCD is a square.
  
  Proof,
  
  In ΔAOB and ΔCOD,
  
  AO = CO (Diagonals bisect each other)
  
  ∠AOB = ∠COD (Vertically opposite)
  
  OB = OD (Diagonals bisect each other)
  
  , ΔAOB ≅ ΔCOD [SAS congruency]
  
  Thus,
  
  AB = CD [CPCT] — (i)
  
  also,
  
  ∠OAB = ∠OCD (Alternate interior angles)
  
  ⇒ AB || CD
  
  Now,
  
  In ΔAOD and ΔCOD,
  
  AO = CO (Diagonals bisect each other)
  
  ∠AOD = ∠COD (Vertically opposite)
  
  OD = OD (Common)
  
  , ΔAOD ≅ ΔCOD [SAS congruency]
  
  Thus,
  
  AD = CD [CPCT] — (ii)
  
  also,
  
  ⇒ AD = BC = CD = AB — (ii)
  
  and ∠ADC+∠BCD = 180° (co-interior angles)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question and a detailed solution to the question. The question asks to prove that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then the quadrilateral is a square. The solution provided uses geometric properties and congruency theorems to prove the statement.
  
  Label: question-and-answer