Geometry essay / paragraph proof

Question

Given: XN is congruent to NZ and NY is congruent to NW.

Based on the information, can you determine that the quadrilateral must be a parallelogram? Explain.

Answer 1

The question is a bit ambiguous as suggested in a comment above. The given congruent segments are consistent with the bisecting diagonals of a parallelogram (figure on left), but they are also consistent with an isosceles trapezoid (figure on right) — hence the ambiguity.

If you have the quadrilateral on the left, then you can put together a nice proof that it's a parallelogram based on SAS congruence of triangles formed by the bisecting diagonals:

The congruent triangles are ΔXYN ≅ ΔZWN and ΔXWN ≅ ΔZYN.

Here's the proof that the quadrilateral on the left is a parallelogram.

XN ≅ NZ and NY ≅ NW are given.

∠XNY ≅ ∠ZNW because they're vertical angles.

So ΔXYN ≅ ΔZWN by the SAS postulate.

∠XNW ≅ ∠ZNY because they're vertical angles.

So ΔXWN ≅ ΔZYN by the SAS postulate.

XY ≅ ZW and XW ≅ ZY because corresponding sides of congruent triangles are congruent.

So opposite sides of quadrilateral WXYZ are congruent.

Therefore quadrilateral WXYZ is a parallelogram.

Answer 2

Well, duh! Especially if you learn to get really involved with the problems or assignments; meaning, review your answer or work the problem another way or insert your solution into the original problem to check your work; that way you can get credit for what you know 100% of the time instead of letting brain farts and oversights count against you.

Answer 3

We aim to prove that this quadrilateral is a parallelogram. One of the properties of a parallelogram is that opposite sides are congruent.

1. Consider Opposite Sides:
2. XN and WY are opposite sides.
3. NZ and YW are opposite sides.
4. Given Congruencies:
5. XN is congruent to NZ (Given)
6. NY is congruent to NW (Given)
7. Proving Congruency of Opposite Sides:
8. We can observe that XN is congruent to NZ, making one pair of opposite sides congruent.
9. Similarly, NY is congruent to NW, making another pair of opposite sides congruent.
10. Conclusion:
11. Since both pairs of opposite sides are congruent (XN is congruent to NZ and NY is congruent to NW), we can conclude that the quadrilateral WNYX is a parallelogram.

So, based on the given information, we can determine that the quadrilateral must be a parallelogram.

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Best regards,

Ben