Question

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

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Answer 1

In parallelogjam ABCD

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))angle ACD+angle BDC =180 °

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))angle ACD+angle BDC =180 °2angleACD =180°

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))angle ACD+angle BDC =180 °2angleACD =180°angle ACD = angle BDC = 90°

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))angle ACD+angle BDC =180 °2angleACD =180°angle ACD = angle BDC = 90°angle ACD=90°

In parallelogjam ABCDangle ABD = angle BDC (AC=BD (GIVEN))angle ACD+angle BDC =180 °2angleACD =180°angle ACD = angle BDC = 90°angle ACD=90°So,Parrallelogram ABCD is a rectangle

OR.

lets say, ABCD is a parallelogram

lets say, ABCD is a parallelogram

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]∴ ΔABD ≅ ΔDCA [SSS congruence criterion]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]∴ ΔABD ≅ ΔDCA [SSS congruence criterion]∠BAD = ∠CDA [CPCT]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]∴ ΔABD ≅ ΔDCA [SSS congruence criterion]∠BAD = ∠CDA [CPCT]∠BAD + ∠CDA = 180° [Adjacent angles of a parallelogram are supplementary.]

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]∴ ΔABD ≅ ΔDCA [SSS congruence criterion]∠BAD = ∠CDA [CPCT]∠BAD + ∠CDA = 180° [Adjacent angles of a parallelogram are supplementary.]So, ∠BAD and ∠CDA are right angles as they are congruent and supplementary.

lets say, ABCD is a parallelogramGiven that the diagonals AC and BD of parallelogram ABCD are equal in length .Consider triangles ABD and ACD.AC = BD [Given]AB = DC [opposite sides of a parallelogram]AD = AD [Common side]∴ ΔABD ≅ ΔDCA [SSS congruence criterion]∠BAD = ∠CDA [CPCT]∠BAD + ∠CDA = 180° [Adjacent angles of a parallelogram are supplementary.]So, ∠BAD and ∠CDA are right angles as they are congruent and supplementary.Therefore, parallelogram ABCD is a rectangle since a parallelogram with one right interior angle is a rectangle.

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