question
choose all answers that describe the polygon vwxy if vw = 12, wx = 12, xy = 12, yv = 12, m∠v = m∠x, and m∠w = m∠y.
answer
parallelogram
quadrilateral
rectangle
rhombus
square
trapezoid

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Accepted Answer

# Brief Explanations:
1. **Quadrilateral**: A polygon with four sides. Since $VWXY$ has four sides ($VW, WX, XY, YV$), it is a quadrilateral.
2. **Parallelogram**: A quadrilateral with both pairs of opposite sides equal and both pairs of opposite angles equal. Here, $VW = XY = 12$ and $WX = YV = 12$ (opposite sides equal), and $m\angle V=m\angle X$, $m\angle W = m\angle Y$ (opposite angles equal), so it is a parallelogram.
3. **Rhombus**: A parallelogram with all four sides equal. Since $VW = WX=XY = YV = 12$, all sides are equal, so it is a rhombus.
4. **Rectangle**: A parallelogram with four right angles. We know opposite angles are equal ($m\angle V=m\angle X$, $m\angle W = m\angle Y$) and the sum of interior angles of a quadrilateral is $360^\circ$. Let $m\angle V = m\angle X = a$ and $m\angle W=m\angle Y = b$. Then $2a + 2b=360^\circ$, or $a + b = 180^\circ$. If we assume the angles are right angles (since in a rhombus with equal opposite angles summing to $180^\circ$ per pair, if all sides are equal and opposite angles equal, it can be a rectangle), but actually, since all sides are equal and it's a parallelogram, if angles are right angles, it's a square, but also a rectangle. Wait, more accurately, since it's a parallelogram with all sides equal (rhombus) and if angles are right angles, it's a square (which is also a rectangle). But even without assuming right angles, the given angle conditions (opposite angles equal) and side conditions (all sides equal) make it a parallelogram, rhombus, and since it's a quadrilateral. Also, a square is a special case of rhombus, rectangle, and parallelogram. Let's check square: A square is a rhombus with four right angles. We know all sides are equal, and if we consider the angle sum: $2a + 2b=360$, and if $a = b$ (since in a rhombus with all sides equal, if opposite angles are equal and let's see, if $a=b$, then $4a = 360$, $a = 90^\circ$), so angles are right angles. So:
 - **Square**: A rhombus with four right angles. Since all sides are equal and angles are right angles (as shown by angle sum: $2a+2b = 360$, and since it's a rhombus, if we assume symmetry, $a = b$, so $a = b=90^\circ$), so it's a square. A square is also a rectangle (a rectangle with all sides equal) and a parallelogram and a rhombus and a quadrilateral.
 - **Trapezoid**: A trapezoid has at least one pair of parallel sides. But a parallelogram has two pairs of parallel sides, so it is a trapezoid? Wait, no, in some definitions, a trapezoid has exactly one pair of parallel sides, but in inclusive definitions, it has at least one. However, since it's a parallelogram (two pairs of parallel sides), it would be a trapezoid in inclusive definition, but the main ones here are quadrilateral, parallelogram, rhombus, square, rectangle. Wait, let's re - evaluate:

Wait, the side lengths: $VW = WX=XY = YV = 12$ (all sides equal), opposite angles equal ($m\angle V=m\angle X$, $m\angle W = m\angle Y$).

- **Quadrilateral**: 4 - sided, so yes.
- **Parallelogram**: Opposite sides equal ($VW = XY$, $WX = YV$) and opposite angles equal ($m\angle V=m\angle X$, $m\angle W = m\angle Y$), so yes.
- **Rhombus**: Parallelogram with all sides equal ($VW = WX=XY = YV$), so yes.
- **Square**: A rhombus with four right angles. Since the sum of interior angles of a quadrilateral is $360^\circ$, and $m\angle V=m\angle X$, $m\angle W = m\angle Y$, let $m\angle V=m\angle X = x$ and $m\angle W=m\angle Y = y$. Then $2x + 2y=360^\circ\Rightarrow x + y = 180^\circ$. In a rhombus, adjacent angles are supplementary. If we consider that in a rhombus with all sides equal, if $x=y$, then $x=y = 90^\circ$, so it's a square. So square is valid.
- **Rectangle**: A parallelogram with four right angles. Since it's a square (which is a special rectangle) or since it's a parallelogram with all sides equal (rhombus) and if angles are right angles (which they are, as shown above), it's a rectangle.
- **Trapezoid**: If we use the inclusive definition (at least one pair of parallel sides), then a parallelogram is a trapezoid. But usually, trapezoid is defined as having exactly one pair of parallel sides, so it's not a trapezoid (since it has two pairs). So trapezoid is not correct.

So the correct ones are Quadrilateral, Parallelogram, Rhombus, Square, Rectangle. Wait, let's re - check:

- Quadrilateral: Yes, 4 sides.
- Parallelogram: Yes, opposite sides equal, opposite angles equal.
- Rhombus: Yes, parallelogram with all sides equal.
- Square: Yes, rhombus with all angles right (since in a rhombus, adjacent angles are supplementary, and if opposite angles are equal, then all angles are equal (since $x=y$ from $x + y=180$ and $x = y$ implies $x = y=90$)).
- Rectangle: Yes, parallelogram with right angles (square is a rectangle).

# Answer:
Quadrilateral, Parallelogram, Rhombus, Square, Rectangle

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