Question

for each of the following quadrilaterals, select all the properties that must be true. (a) rectangle (b) trapezoid (c) parallelogram all sides congruent two pairs of parallel sides only one pair of parallel sides four right angles

Explanation:

Step 1: Recall properties of each quadrilateral

• Rectangle: A rectangle is a parallelogram with four right angles. Opposite sides are congruent and parallel (two pairs of parallel sides). All angles are right angles (four right angles). But not all sides are congruent (only opposite sides are congruent in a general rectangle; a square is a special rectangle with all sides congruent).
• Trapezoid: A trapezoid (in the US definition) has only one pair of parallel sides. The sides are not necessarily congruent, and it does not have four right angles (unless it's a right trapezoid, but that's a special case, not a property that must be true for all trapezoids).
• Parallelogram: A parallelogram has two pairs of parallel sides. Opposite sides are congruent, but not all sides are necessarily congruent (a rhombus is a special parallelogram with all sides congruent). It does not have four right angles (unless it's a rectangle, a special parallelogram).

Step 2: Match properties to each quadrilateral

(a) Rectangle

• All sides congruent: No (only opposite sides are congruent in general rectangle).
• Two pairs of parallel sides: Yes (by definition of rectangle as a parallelogram).
• Only one pair of parallel sides: No (has two pairs).
• Four right angles: Yes (by definition).

(b) Trapezoid

• All sides congruent: No (sides are not necessarily congruent).
• Two pairs of parallel sides: No (has only one pair).
• Only one pair of parallel sides: Yes (by definition of trapezoid).
• Four right angles: No (only right trapezoid has right angles, not all trapezoids).

(c) Parallelogram

• All sides congruent: No (only opposite sides are congruent in general parallelogram).
• Two pairs of parallel sides: Yes (by definition).
• Only one pair of parallel sides: No (has two pairs).
• Four right angles: No (only rectangles, a special parallelogram, have four right angles).

Answer:

Quadrilateral	All sides congruent	Two pairs of parallel sides	Only one pair of parallel sides	Four right angles
(b) Trapezoid	$\square$ No	$\square$ No	$\boldsymbol{\checkmark}$ Yes	$\square$ No
(c) Parallelogram	$\square$ No	$\boldsymbol{\checkmark}$ Yes	$\square$ No	$\square$ No

(Note: In the table, mark the boxes with $\checkmark$ for the properties that hold for each quadrilateral.)