Let's analyze each of the three quadrilaterals (Square, Parallelogram, and Trapezoid) and identify which properties must be true for each one. We'll use the properties provided to determine the appropriate selections.
1. Square
- All sides congruent: True (A defining property of a square is that all four sides are of equal length.)
- Four right angles: True (A square has four 90-degree angles.)
- Only one pair of parallel sides: False (A square has two pairs of parallel sides.)
- Two pairs of parallel sides: True (Both pairs of opposite sides in a square are parallel.)
So, for a square:
- All sides congruent: [tex]$\square$[/tex]
- Four right angles: [tex]$\square$[/tex]
- Only one pair of parallel sides:
- Two pairs of parallel sides: [tex]$\square$[/tex]
2. Parallelogram
- All sides congruent: False (A parallelogram typically has opposite sides of equal length, not necessarily all four sides unless it is a rhombus.)
- Four right angles: False (This property is true only for a rectangle, which is a specific type of parallelogram.)
- Only one pair of parallel sides: False (A parallelogram by definition has two pairs of opposite sides that are parallel.)
- Two pairs of parallel sides: True (As mentioned, a parallelogram has two pairs of parallel sides.)
So, for a parallelogram:
- All sides congruent: [tex]$\square$[/tex]
- Four right angles: [tex]$\square$[/tex]
- Only one pair of parallel sides: [tex]$\square$[/tex]
- Two pairs of parallel sides: [tex]$\square$[/tex]
3. Trapezoid
- All sides congruent: False (A trapezoid does not require all sides to be of equal length.)
- Four right angles: False (A trapezoid does not require any right angles, although it can have one or two in the case of a right trapezoid.)
- Only one pair of parallel sides: True (A trapezoid is defined as having exactly one pair of parallel sides.)
- Two pairs of parallel sides: False (A trapezoid, by definition, cannot have both pairs of sides parallel.)
So, for a trapezoid:
- All sides congruent: [tex]$\square$[/tex]
- Four right angles: [tex]$\square$[/tex]
- Only one pair of parallel sides: [tex]$\square$[/tex]
- Two pairs of parallel sides: [tex]$\square$[/tex]
Thus, the filled-in table should be:
[tex]\[
\begin{tabular}{|l|c|c|c|c|}
\hline
& All sides congruent & Four right angles & Only one pair of parallel sides & Two pairs of parallel sides \\
\hline
(a) Square & \square & \square & & \square \\
\hline
(b) Parallelogram & & & & \square \\
\hline
(c) Trapezoid & & & \square & \\
\hline
\end{tabular}
\][/tex]
