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Categories of quadrilaterals based on sides and angles.

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All squares are rectangles, but not all rectangles are squares. How is this possible?

A quadrilateral is a polygon with four sides. There are many common special quadrilaterals that you should be familiar with. Below, these special quadrilaterals are described with their definitions and some properties. Additional properties are explored further and proved in another concept.

• A kite is a convex quadrilateral with two pairs of adjacent congruent sides such that not all sides are congruent. The word adjacent means “next to”, so the congruent sides are next to each other. Note: some texts leave out the stipulation that “not all sides are congruent”. If this is the case, it is possible for a kite to be a rhombus if all four sides are congruent. To note that sides are congruent on a picture, draw corresponding tick marks through the middles of the congruent sides.

• A trapezoid is a quadrilateral with exactly one pair of parallel sides. Note: some texts leave out the word “exactly”, which means quadrilaterals with two pairs of parallel sides are sometimes considered trapezoids. Here, assume trapezoids have exactly one pair of parallel sides. To note that sides are parallel on a picture, draw corresponding arrow markings through the middles of the parallel sides.

• An isosceles trapezoid is a trapezoid with one pair of congruent sides. An additional property of isosceles trapezoids is base angles are congruent.

• A parallelogram is a quadrilateral with two pairs of parallel sides. Some additional properties of parallelograms are opposite sides are congruent and opposite angles are congruent.

• A rectangle is a quadrilateral with four right angles. All rectangles are parallelograms. An additional property of rectangles is diagonals are congruent.

• A rhombus is a quadrilateral with four congruent sides. All rhombuses are parallelograms.

• A square is a quadrilateral with four right angles and four congruent sides. All squares are rectangles and rhombuses.

Notice that these categories of quadrilaterals overlap. A square is not only a square, but also a rhombus, a rectangle, a parallelogram, and a quadrilateral. This means that a square will have all the same properties as rhombuses, rectangles, parallelograms, and quadrilaterals.

The following diagram shows the hierarchy of quadrilaterals.

1. What types of quadrilaterals have four right angles?

Rectangles and squares have four right angles.

2. A quadrilateral has two pairs of parallel sides. What type of quadrilateral must it be? What type of quadrilateral could it be?

Two pairs of parallel sides is the definition of a parallelogram, so this quadrilateral must be a parallelogram. It could be a rectangle, a square, or a rhombus if it satisfied the definition of any of those quadrilaterals.

Solving for Unknown Values

Solve for \begin{align*}x\end{align*} (picture not drawn to scale).

This quadrilateral is marked as having four congruent sides, so it is a rhombus. Rhombuses are parallelograms, so they have all the same properties as parallelograms. One property of parallelograms is that opposite angles are congruent. This means that the marked angles in this rhombus must be congruent.

\begin{align*}x+7 &=2x \\ x &=7\end{align*}

Examples

Example 1

Earlier, you were asked how it's possible that all squares are rectangles, but not all rectangles are squares.

Rectangles are defined as quadrilaterals with four right angles. Squares are defined as quadrilaterals with four right angles and four congruent sides. Because all squares have four right angles and satisfy the definition for rectangles, they can all also be called rectangles. On the other hand, not all rectangles have four congruent sides, so not all rectangles can also be called squares.

Example 2

Draw a square. Draw in the diagonals of the square. Make at least one conjecture about the diagonals of the square.

To make a conjecture means to make an educated guess. There are a few conjectures you might make about the diagonals of a square. These conjectures will be proved in a later concept. Here are some possible conjectures:

• diagonals of a square are congruent
• diagonals of a square are perpendicular
• diagonals of a square bisect each other (cut each other in half)
• diagonals of a square bisect the angles (cut the \begin{align*}90^\circ\end{align*} angles in half)

Example 3

A quadrilateral has four congruent sides. What type of quadrilateral must it be? What type of quadrilateral could it be?

It must be a rhombus and therefore also a parallelogram. It could be a square.

Example 4

Solve for \begin{align*}x\end{align*} (picture not drawn to scale).

This is a parallelogram so opposite sides are congruent.

\begin{align*}3x+1 &=5x-12 \\ 2x &=13 \\ x &=6.5\end{align*}

Review

Decide whether each statement is always, sometimes, or never true. Explain your answer.

1. A square is a rectangle.

2. A rhombus is a square.

3. An isosceles trapezoid is a trapezoid.

4. A parallelogram is a quadrilateral.

5. A square is a parallelogram.

6. A trapezoid is a parallelogram.

Decide what type of quadrilateral it must be and what type of quadrilateral it could be based on the description.

7. A quadrilateral has 4 congruent angles.

8. A quadrilateral has 2 pairs of congruent sides.

9. Draw a kite. Draw in its diagonals. Make at least one conjecture about the diagonals of kites.

10. Draw a rectangle. Draw in its diagonals. Make at least one conjecture about the diagonals of rectangles.

11. Draw a rhombus. Draw in its diagonals. Make at least one conjecture about the diagonals of rhombuses.

12. Draw a kite. Make a conjecture about the opposite angles of kites.

Use the markings on the shapes below to identify the shape. Then, solve for \begin{align*}x\end{align*}. Note: pictures are not drawn to scale.

13.

14.

15. Make a conjecture about the adjacent angles of a parallelogram (such as the ones marked in the picture below). How must they be related?

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Color Highlighted Text Notes

Vocabulary Language: English

Kite

Parallelogram

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

A quadrilateral is a closed figure with four sides and four vertices.

Rectangle

A rectangle is a quadrilateral with four right angles.

Rhombus

A rhombus is a quadrilateral with four congruent sides.

Trapezoid

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