Edwidge draws parallelogram STUV with a center at O. What must be true to prove that STUV is a parallelogram?

Answer 1

To prove that quadrilateral STUV is a parallelogram, one of the following conditions must be satisfied:

1. Both pairs of opposite sides are parallel: If \( ST \parallel UV \) and \( SU \parallel TV \), then STUV is a parallelogram.

2. Both pairs of opposite sides are equal in length: If \( ST = UV \) and \( SU = TV \), then STUV is a parallelogram.

3. One pair of opposite sides is both parallel and equal: If \( ST \parallel UV \) and \( ST = UV \), or \( SU \parallel TV \) and \( SU = TV \), then STUV is a parallelogram.

4. Diagonals bisect each other: If the diagonals (SU and TV) bisect each other at point O, then STUV is a parallelogram.

5. Using angle properties: If one pair of opposite angles are equal, then the quadrilateral is a parallelogram. For example, if \( \angle S = \angle U \) and \( \angle T = \angle V \), then STUV is a parallelogram.

Any one of these conditions is sufficient to prove that STUV is a parallelogram.

Answer 2

The answers are,
A: ST ≅ TU and UV ≅ SV
B: ST ≅ UV and TU ≅ SV
C: SO ≅ VO and TO ≅ UO
D: SO ≅ TO and UO ≅ VO

Answer 3

To determine which provided statement is true to prove that STUV is a parallelogram, it's important to focus on the properties of a parallelogram regarding its sides and diagonals.

Given the options:

A: \( ST \cong TU \) and \( UV \cong SV \)
This option states that opposite sides are equal. However, it suggests side lengths that do not correspond to a typical parallelogram configuration.

B: \( ST \cong UV \) and \( TU \cong SV \)
This option indicates that both pairs of opposite sides are equal. This is one of the conditions (specifically the second one) that proves a quadrilateral is a parallelogram.

C: \( SO \cong VO \) and \( TO \cong UO \)
This option suggests that diagonals are bisected, and since O is the center, this would imply that the diagonals of the quadrilateral bisect each other, which is also a property of parallelograms.

D: \( SO \cong TO \) and \( UO \cong VO \)
Similar to option C, this also suggests that the diagonals bisect each other.

Both options B and C demonstrate properties that would prove STUV is a parallelogram. However, if we consider which one aligns directly with common theorems, B is the cleaner representation of the necessary conditions relating to sides.

In summary, when referring to conditions that prove a quadrilateral is a parallelogram, B and C both hold merit, but B explicitly asserts the equality of opposite sides, while C refers to diagonal bisection. This makes B the most directly applicable for proving that STUV is a parallelogram through side equality.

Thus, with respect to typical proof structures, B (ST ≅ UV and TU ≅ SV) is the most straightforward answer to prove that STUV is a parallelogram based on the properties of its sides.