Question

choose all answers that describe the quadrilateral efgh if \\(\overline{ef} \parallel \overline{gh}, \overline{ef} \cong \overline{gh}\\), and diagonals have equal length: \\(eg = fh\\).

answer attempt 1 out of 2

\\(\square\\) parallelogram
\\(\square\\) rectangle
\\(\square\\) rhombus
\\(\square\\) square
\\(\square\\) trapezoid

Explanation:

Step1: Analyze Parallelogram

A quadrilateral with one pair of opposite sides parallel and congruent ($\overline{EF} \parallel \overline{GH}$, $\overline{EF} \cong \overline{GH}$) is a parallelogram. So EFGH is a parallelogram.

Step2: Analyze Rectangle

In a parallelogram, if diagonals are equal ($EG = FH$), then it is a rectangle. Since EFGH is a parallelogram with equal diagonals, it is a rectangle.

Step3: Analyze Rhombus

A rhombus requires all sides congruent or diagonals perpendicular, which is not given here. So not a rhombus.

Step4: Analyze Square

A square is a rhombus and a rectangle, but we don't know if all sides are equal, so not necessarily a square.

Step5: Analyze Trapezoid

A parallelogram is a trapezoid (inclusive definition), but the more specific descriptions (parallelogram, rectangle) are better. However, since the question says "ALL answers", we check: A trapezoid has at least one pair of parallel sides. EFGH has one pair (and is a parallelogram, so two pairs), so it is a trapezoid (in inclusive sense). But also, parallelogram and rectangle are more specific. Wait, but let's re - check:
- Parallelogram: Yes, as one pair of opposite sides parallel and congruent.
- Rectangle: Yes, because parallelogram with equal diagonals.
- Trapezoid: In the inclusive definition (at least one pair of parallel sides), a parallelogram is a trapezoid. But some definitions of trapezoid are exclusive (exactly one pair), but in many modern definitions, it's inclusive. However, the key here is that the given conditions first give a parallelogram, then with equal diagonals, a rectangle. Also, a square is a rectangle with all sides equal, but we don't have info about side lengths, so square is not necessarily. Rhombus is out. So the correct ones are Parallelogram, Rectangle, and (in inclusive trapezoid definition) Trapezoid. But let's check standard definitions:
- Parallelogram: A quadrilateral with both pairs of opposite sides parallel (which we have, since one pair is parallel and congruent, so the other pair must be too).
- Rectangle: A parallelogram with equal diagonals (which we have).
- Trapezoid: If we use the inclusive definition (at least one pair of parallel sides), then yes. But sometimes trapezoid is defined as exactly one pair, but in that case, a parallelogram (with two pairs) would not be a trapezoid. However, in the context of this problem, since we have a parallelogram, the main ones are Parallelogram and Rectangle, and also, a square is a rectangle with all sides equal, but we don't know if all sides are equal, so square is not. Rhombus is out. Wait, maybe the problem uses the inclusive trapezoid definition. But let's go back to the properties:
Given $\overline{EF} \parallel \overline{GH}$ and $\overline{EF} \cong \overline{GH}$, so by the theorem, EFGH is a parallelogram (a quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram). Then, in a parallelogram, if diagonals are equal ($EG = FH$), then it is a rectangle. Also, a parallelogram is a trapezoid (in inclusive definition). But let's check the options:
- Parallelogram: Correct.
- Rectangle: Correct (since parallelogram with equal diagonals).
- Trapezoid: Correct (in inclusive definition, as it has parallel sides).
- Rhombus: Incorrect, no info about all sides equal.
- Square: Incorrect, no info about all sides equal.

Wait, but maybe the problem's definition of trapezoid is exclusive. But in many math curricula, especially in the US, the inclusive definition is becoming more common. However, let's re - evaluate:

The key properties:
1. $\overline{EF} \parallel \overline{GH}$ and $\overline{EF} \cong \overline{GH}$: This implies EFGH is a parallelogram (theorem: if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram).
2. Diagonals $EG = FH$: In a parallelogram, if diagonals are equal, then the parallelogram is a rectangle (theorem: in a parallelogram, equal diagonals imply it is a rectangle).

Now, about trapezoid: A trapezoid is a quadrilateral with at least one pair of parallel sides. Since EFGH is a parallelogram, it has two pairs of parallel sides, so it satisfies the "at least one" condition (in inclusive definition), so it is a trapezoid.

Now, rhombus: A rhombus has all sides congruent. We only know one pair of sides is congruent (and since it's a parallelogram, the other pair is too), but we don't know if adjacent sides are congruent. So not a rhombus.

Square: A square is a rhombus and a rectangle, so it needs all sides congruent and all angles right angles. We know it's a rectangle (right angles, since it's a rectangle), but we don't know if all sides are congruent, so not necessarily a square.

So the correct answers are Parallelogram, Rectangle, and Trapezoid (in inclusive definition). But let's check the options again. Wait, maybe the trapezoid here is considered as a non - parallelogram trapezoid (exclusive definition), but in that case, a parallelogram is not a trapezoid. However, in the problem's context, since it's a math problem (probably using inclusive trapezoid definition or at least considering that a parallelogram is a trapezoid), but the main ones are Parallelogram and Rectangle, and also, since the question says "ALL answers", we need to check each:

- Parallelogram: Yes.
- Rectangle: Yes.
- Rhombus: No.
- Square: No.
- Trapezoid: Yes (in inclusive definition).

But maybe in the problem's intended answer, trapezoid is not considered (using exclusive definition), but let's go with the theorems:

1. Parallelogram: Yes, because one pair of opposite sides is parallel and congruent.
2. Rectangle: Yes, because parallelogram with equal diagonals.
3. Trapezoid: Yes (in inclusive definition, as it has at least one pair of parallel sides).

But maybe the problem considers trapezoid as having exactly one pair, so in that case, trapezoid is not correct. But the majority of modern math definitions (especially in Common Core) use the inclusive definition for trapezoid.

Answer:

A. Parallelogram, B. Rectangle, E. Trapezoid