Proving a Quadrilateral with Congruent Diagonals as a Parallelogram: Side-Side-Side and Side-Angle-Side Congruence Criteria

A quadrilateral with congruent diagonals is a parallelogram

To prove that a quadrilateral with congruent diagonals is a parallelogram, we need to show that opposite sides are congruent and opposite angles are congruent

To prove that a quadrilateral with congruent diagonals is a parallelogram, we need to show that opposite sides are congruent and opposite angles are congruent.

Let’s consider the given quadrilateral ABCD, with diagonals AC and BD intersecting at point E.

To prove that opposite sides are congruent, we can use the Side-Side-Side (SSS) congruence criterion. Let’s prove that AB and CD are congruent.

In ΔABC and ΔCDA:
– Side AC is common to both triangles.
– Diagonals AC and BD are congruent (given).
– Side BC is congruent to side AD as they are opposite sides of a quadrilateral.

According to the SSS congruence criterion, ΔABC is congruent to ΔCDA.

Therefore, we can conclude that AB is congruent to CD.

Next, to prove that opposite angles are congruent, we can use the Side-Angle-Side (SAS) congruence criterion. Let’s prove that ∠A and ∠C are congruent.

In ΔABC and ΔCDA:
– We have already proved that side AC is congruent in the previous step.
– Diagonals AC and BD are congruent (given).
– Side BC is congruent to side AD as they are opposite sides of a quadrilateral.

According to the SAS congruence criterion, ΔABC is congruent to ΔCDA.

Therefore, we can conclude that ∠A is congruent to ∠C.

Since opposite sides and opposite angles are congruent, we can say that the quadrilateral ABCD is a parallelogram.

Hence, we have proved that a quadrilateral with congruent diagonals is a parallelogram.

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