Lesson 31: Investigation 3: Classifying Quadrilaterals

New Concept

Quadrilaterals are four-sided polygons. They are classified into a family of shapes based on key properties like parallel sides, side lengths, and angles.

What’s next

Next, you'll use a Venn diagram to visualize these shape families and then explore their unique properties, including angles and symmetry.

Property

A quadrilateral with two pairs of parallel sides is called a parallelogram.

Examples

• A rhombus is a parallelogram where all four sides are equal in length.
• A square is a parallelogram where all sides are equal and all angles are $90^\circ$.
• A rectangle is a parallelogram where all angles are $90^\circ$.

Explanation

Think of a rectangle that got a little push. A parallelogram has two sets of parallel sides, which means opposite sides are always equal in length, and opposite angles are always identical. This balanced setup makes it the parent shape for other cool quadrilaterals like squares, rectangles, and rhombuses, which are all just fancy parallelograms with extra rules.

Property

A quadrilateral with four right angles is a special kind of parallelogram called a rectangle.

Examples

• A rectangle with sides of length 4 cm and 9 cm has a perimeter of $2(4+9) = 26$ cm.
• A square with a side length of 6 is a special type of rectangle because all its angles are $90^\circ$.
• Transforming a rectangle by shifting its sides creates a parallelogram that is no longer a rectangle.

Explanation

Meet the straight-laced member of the parallelogram family! A rectangle is a parallelogram that is committed to perfect posture, with four sharp $90^\circ$ right angles. Its opposite sides are parallel and equal, but the strict corner rule makes it super stable and predictable. It is the reliable shape you see in doors, phone screens, and notebooks everywhere.

Property

A quadrilateral with one pair of parallel sides is a trapezoid.

Examples

• In a trapezoid, the two parallel bases can have lengths of 5 cm and 10 cm, while the non-parallel legs can be 6 cm and 7 cm.
• An isosceles trapezoid has one pair of parallel sides and its non-parallel sides are equal in length.
• Figure E from the lesson is a trapezoid because it has exactly one pair of parallel sides.

Explanation

This shape is all about having just one set of parallel sides, called bases. Think of it as a quadrilateral that couldn't fully commit to being a parallelogram. The other two sides, the legs, are not parallel and can be different lengths. If the legs happen to be equal, you have a special, more symmetrical version called an isosceles trapezoid.