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Use your notes from the "Work Backwards to Prove" activity to complete the proof below that if the diagonals of a parallelogram are congruent, then that parallelogram must be a rectangle.

**Given:**
- ABCD is a parallelogram with AB parallel to CD and AD parallel to BC.
- Diagonal AC is congruent to diagonal BD.

**Prove:**
- ABCD is a rectangle (angles A, B, C, and D are right angles).

1. I know angle 1 is congruent to angle 2 because they represent the same segment.
2. I know angle 3 is congruent to angle 14 because it's given.
3. I know angle 5 is congruent to angle 6 because ABCD is a parallelogram (given), and opposite sides of a parallelogram are congruent.
4. Because angle 8 is congruent to angle 9, angle 10 is congruent to angle 11, and angle 12 is congruent to angle 13 by the Side-Side-Side Triangle Congruence Theorem, triangles 14 and 15 are congruent.
5. Angle 16 is congruent to angle 17 because they are corresponding parts of two congruent triangles.
6. Angles 18 and 19 are right angles because they're congruent and supplementary (since they are adjacent angles in a parallelogram). Congruent supplementary angles must be right angles.
7. Opposite angles in a parallelogram are congruent, so if angles 20 and 21 are right angles, then angles 22 and 23 must be right angles too.
8. I know ABCD is a rectangle because angles 25, 26, 27, and 28 are all right angles, and a quadrilateral with four right angles is a rectangle.

Answer :

If the diagonals of a parallelogram are congruent, then the parallelogram must be a rectangle because the opposite angles of a parallelogram are congruent and the opposite sides of a parallelogram are congruent and parallel.

Proof that if the diagonals of a parallelogram are congruent, then that parallelogram must be a rectangle:

Given: ABCD is a parallelogram with AB parallel to CD and AD parallel to BC. Diagonal AC is congruent to Diagonal BD.

Prove: ABCD is a rectangle (Angles A, B, C, and D are right angles).

Proof:

Since ABCD is a parallelogram, we know that opposite sides are congruent and parallel. Therefore, AB is congruent to CD and AD is congruent to BC.

Since Diagonal AC is congruent to Diagonal BD, we know that triangles ABC and CDA are congruent by the SSS (Side-Side-Side) Congruency Theorem.

By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Theorem, we know that Angle A is congruent to Angle C.

Since AB is parallel to CD, we know that Angle A and Angle C are alternate interior angles. Therefore, Angle A and Angle C are congruent.

Since AD is parallel to BC, we know that Angle B and Angle D are alternate interior angles. Therefore, Angle B and Angle D are congruent.

Since ABCD is a parallelogram, we know that opposite angles are congruent. Therefore, Angle A is congruent to Angle C and Angle B is congruent to Angle D.

Since Angle A is congruent to Angle C and Angle B is congruent to Angle D, we know that all four angles of ABCD are congruent.

A quadrilateral with all four angles congruent is a rectangle.

Therefore, ABCD is a rectangle.

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The question probable may be:

Use your notes from the forward to backward activity to prove that... if the diagonals of a parallelogram are congruent, then that parallelogram must be a rectangle. Given: ABCD is a parallelogram with AB parallel to CD and AD parallel to BC. Diagonal AC is congruent to Diagonal BD. Prove: ABCD is a rectangle (Angles A, B, C, and D are right angles).

Thank you for reading the article Use your notes from the Work Backwards to Prove activity to complete the proof below that if the diagonals of a parallelogram are congruent then. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!

Rewritten by : Jeany

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