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If the diagonals of a parallelogram are equal, then show that it is a rectangle.
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If the diagonals of a parallelogram are equal, then show that it is a ...
If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle. This can be proven using the properties of a rectangle and the properties of a parallelogram.
A rectangle is a four-sided polygon with opposite sides that are equal in length and opposite angles that are equal in measure. If the diagonals of a parallelogram are equal, then the opposite angles of the parallelogram are equal in measure.
In a parallelogram, the opposite angles are always equal in measure, regardless of whether the diagonals are equal or not. Therefore, if the diagonals of a parallelogram are equal, then the opposite angles of the parallelogram must also be equal in measure.
Since a rectangle is a parallelogram with opposite sides that are equal in length and opposite angles that are equal in measure, it follows that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.
Therefore, if the diagonals of a parallelogram are equal, then it is a rectangle.
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Understanding the Problem
To prove that a parallelogram with equal diagonals is a rectangle, we will use basic properties of parallelograms and triangles.
Properties of Parallelograms
- A parallelogram is a quadrilateral with opposite sides that are equal and parallel.
- The diagonals of a parallelogram bisect each other.
Equal Diagonals
- Let ABCD be a parallelogram where diagonals AC and BD are equal, i.e., AC = BD.
- Since the diagonals bisect each other, let O be the point of intersection. Therefore, AO = OC and BO = OD.
Triangles and Congruence
- In triangles AOB and COD:
- AO = OC (Diagonals bisect each other)
- BO = OD (Diagonals bisect each other)
- AC = BD (Given)
- By the Side-Side-Side (SSS) congruence criterion, triangles AOB and COD are congruent.
Implication of Congruence
- Since triangles AOB and COD are congruent, we have:
- Angle AOB = Angle COD
- Angle OAB = Angle OCD
Conclusion: Right Angles
- In a parallelogram, if the diagonals are equal, then the angles AOB and COD must be right angles.
- Therefore, all angles in the parallelogram ABCD are right angles.
Final Statement
Since a parallelogram with four right angles is defined as a rectangle, we conclude that if the diagonals of a parallelogram are equal, then it is indeed a rectangle.
To prove that a parallelogram with equal diagonals is a rectangle, we will use basic properties of parallelograms and triangles.
Properties of Parallelograms
- A parallelogram is a quadrilateral with opposite sides that are equal and parallel.
- The diagonals of a parallelogram bisect each other.
Equal Diagonals
- Let ABCD be a parallelogram where diagonals AC and BD are equal, i.e., AC = BD.
- Since the diagonals bisect each other, let O be the point of intersection. Therefore, AO = OC and BO = OD.
Triangles and Congruence
- In triangles AOB and COD:
- AO = OC (Diagonals bisect each other)
- BO = OD (Diagonals bisect each other)
- AC = BD (Given)
- By the Side-Side-Side (SSS) congruence criterion, triangles AOB and COD are congruent.
Implication of Congruence
- Since triangles AOB and COD are congruent, we have:
- Angle AOB = Angle COD
- Angle OAB = Angle OCD
Conclusion: Right Angles
- In a parallelogram, if the diagonals are equal, then the angles AOB and COD must be right angles.
- Therefore, all angles in the parallelogram ABCD are right angles.
Final Statement
Since a parallelogram with four right angles is defined as a rectangle, we conclude that if the diagonals of a parallelogram are equal, then it is indeed a rectangle.
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If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?
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If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?.
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a parallelogram are equal, then show that it is a rectangle. Related: Ex 8.1 NCERT Solutions: Quadrilaterals?.
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