Question

end-of-topic review
available online

1. a topic summary reviews the main concepts for the topic
2. a video of the worked example is provided.

practice concepts you learned in composing and decomposing shapes.
9 identify the quadrilaterals that match each description.
a at least one pair of opposite sides is parallel
b at least one pair of opposite angles is congruent

Explanation:

Step1: Recall Quadrilateral Properties

• Parallelograms (including rectangles, rhombuses, squares) have both pairs of opposite sides parallel. Trapezoids (in some definitions) have at least one pair of parallel sides. So for part (a), trapezoids (if defined as at least one pair of parallel sides), parallelograms, rectangles, rhombuses, squares.
• For part (b), parallelograms have opposite angles congruent. Also, isosceles trapezoids have base angles congruent (and opposite angles supplementary, but in some cases, if we consider "at least one pair", parallelograms (all opposite angles congruent), isosceles trapezoids (base angles congruent, and also the pair of base angles: in isosceles trapezoid, $\angle A \cong \angle D$, $\angle B \cong \angle C$, and also $\angle A + \angle B = 180^\circ$, but if we consider "at least one pair", parallelograms (opposite angles congruent) and isosceles trapezoids (base angles congruent, and also the non - adjacent angles? Wait, in isosceles trapezoid, $\angle A \cong \angle D$ and $\angle B \cong \angle C$. So for "at least one pair", parallelograms (opposite angles congruent), isosceles trapezoids (base angles congruent), and also any quadrilateral with one pair of opposite angles congruent (though such quadrilaterals are less common, but by definition, if at least one pair, then parallelograms (since in parallelogram both pairs are congruent), isosceles trapezoids (base angles congruent).

Step2: Analyze Each Part

Part (a)

• Trapezoid (by definition, a quadrilateral with at least one pair of parallel sides in some definitions; in exclusive definition, exactly one pair, but the problem says "at least one", so trapezoid (in inclusive definition), parallelogram, rectangle, rhombus, square.
• Parallelogram: both pairs of opposite sides parallel, so satisfies "at least one".
• Rectangle: special parallelogram, so satisfies.
• Rhombus: special parallelogram, so satisfies.
• Square: special parallelogram, so satisfies.

Part (b)

• Parallelogram: opposite angles are congruent (both pairs), so satisfies.
• Isosceles trapezoid: base angles are congruent ( $\angle A \cong \angle D$ and $\angle B \cong \angle C$), so satisfies "at least one pair".
• Also, a quadrilateral with, say, $\angle A=\angle C$ (even if not a parallelogram or isosceles trapezoid) would satisfy, but the common ones are parallelograms and isosceles trapezoids.

Answer:

Part (a)

Trapezoid (inclusive definition), Parallelogram, Rectangle, Rhombus, Square

Part (b)

Parallelogram, Isosceles Trapezoid (and other quadrilaterals with at least one pair of opposite angles congruent)

(Note: If using the exclusive trapezoid definition (exactly one pair of parallel sides), then trapezoid (exclusive) and parallelogram, rectangle, rhombus, square for part (a). The answer may vary slightly based on the definition of trapezoid used, but the key is to identify quadrilaterals with the given properties.)