Review of Quadrilaterals

The prefix "quad-" means "four", and "lateral" is derived from the Latin word for "side". So a quadrilateral is a four-sided polygon. Since it is a polygon, you know that it is a two-dimensional figure made up of straight sides. A quadrilateral also has four angles formed by its four sides.

 

$\begin{align*}AB\end{align*}$, $\begin{align*}BC\end{align*}$, $\begin{align*}CD\end{align*}$ and $\begin{align*}DA\end{align*}$ are the sides and $\begin{align*}A\end{align*}$, $\begin{align*}B\end{align*}$, $\begin{align*}C\end{align*}$ and $\begin{align*}D\end{align*}$ are the vertices of the quadrilaterals.

Line segments $\begin{align*}AC\end{align*}$ and $\begin{align*}BD\end{align*}$ joining two non-consecutive vertices are called diagonals.

Two sides like $\begin{align*}AB\end{align*}$ and $\begin{align*}AD\end{align*}$ having a common endpoint are called adjacent 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.

Source: CK-12, https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/1.5/primary/lesson/quadrilaterals-geo-ccss/

 This work is licensed under CK-12 Curriculum Materials License.

A kite is a convex quadrilateral with two pairs of adjacent congruent sides such that not all sides are congruent.

The angles between the congruent sides are called vertex angles. The other angles are called non-vertex angles.

The Properties of a Kite:

1. Two pairs of adjacent sides are congruent, i.e., $\begin{align*}AB = AD\end{align*}$ and $\begin{align*}BC = CD\end{align*}$.

2. Non-vertex angles are congruent, i.e., $\begin{align*}\angle{ABC} = \angle{ADC}\end{align*}$.

3. Diagonals intersect each other at right angles, i.e., $\begin{align*}\angle AOB\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle BOC\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle COD\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle DOA\end{align*}$$\begin{align*}= 90^\circ\end{align*}$.

4. The longer diagonal of a kite bisects the shorter one, i.e., $\begin{align*}BO = OD\end{align*}$.

5. The diagonal through the vertex angles is the angle bisector for both angles, i.e., $\begin{align*}\angle{BAC} = \angle{DAC}\end{align*}$ and $\begin{align*}\angle{BCA} = \angle{DCA}\end{align*}$.

6. One of the diagonals bisects the kite, i.e., divides it into two congruent triangles, i.e., $\begin{align*}\triangle ABC \cong \triangle ADC\end{align*}$.

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

 

The Properties of a Trapezoid:

1. One pair of opposite sides are parallel, i.e., $\begin{align*}AB \parallel CD\end{align*}$.

2. The two pairs of adjacent angles along the sides are supplementary, i.e., $\begin{align*}\angle{ABC} + \angle{BCD} = 180^\circ\end{align*}$and $\begin{align*}\angle{CDA} + \angle{DAB} = 180^\circ\end{align*}$.

Note: Some texts leave out the word "exactly", which means quadrilaterals with two pairs of parallel sides are sometimes considered trapezoids. In this course, assume trapezoids have exactly one pair of parallel sides.

An isosceles trapezoid is a trapezoid with the non-parallel sides congruent. An additional property of isosceles trapezoids is base angles are congruent.

 

The Properties of an Isosceles Trapezoid:

1. One pair of opposite sides are parallel, i.e., $\begin{align*}AB \parallel CD\end{align*}$.

2. Two pairs of adjacent angles are supplementary, i.e., $\begin{align*}\angle{ABC} + \angle{BCD} = 180^\circ\end{align*}$and $\begin{align*}\angle{CDA} + \angle{DAB} = 180^\circ\end{align*}$.

3. Base angles are congruent, i.e., $\begin{align*}\angle{ABC} = \angle{DAB}\end{align*}$ and $\begin{align*}\angle{BCD} = \angle{CDA}\end{align*}$.

4. The diagonals are congruent, i.e., $\begin{align*}AC = BD\end{align*}$.

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

The Properties of a Parallelogram:

1. Opposite sides are parallel, i.e., $\begin{align*}AB \parallel CD\end{align*}$ and $\begin{align*}BC \parallel DA\end{align*}$.

2. Opposite sides are congruent, i.e., $\begin{align*}AB = CD\end{align*}$ and $\begin{align*}BC = DA\end{align*}$.

3. Opposite angles are congruent, i.e., $\begin{align*}\angle{ABC} = \angle{CDA}\end{align*}$ and $\begin{align*}\angle{DAB} = \angle{BCD}\end{align*}$.

4. Adjacent angles are supplementary, i.e., $\begin{align*}\angle{ABC} + \angle{BCD} = 180^\circ\end{align*}$, $\begin{align*}\angle{BCD} + \angle{CDA} = 180^\circ\end{align*}$, $\begin{align*}\angle{CDA} + \angle{DAB} = 180^\circ\end{align*}$and $\begin{align*}\angle{DAB} + \angle{ABC} = 180^\circ\end{align*}$.

5. Diagonals bisect each other, i.e., $\begin{align*}BO = OD\end{align*}$ and $\begin{align*}AO = OC\end{align*}$.

6. Each diagonal bisects the parallelogram, i.e., divides it into two congruent triangles $\begin{align*}\triangle ABC \cong \triangle ADC\end{align*}$ and $\begin{align*}\triangle BCD \cong \triangle BAD\end{align*}$.

A rectangle is a quadrilateral with four right angles. All rectangles are parallelograms.

 

The Properties of a Rectangle:

1. Opposite sides are parallel, i.e., $\begin{align*}AB \parallel CD\end{align*}$ and $\begin{align*}BC \parallel DA\end{align*}$.

2. Opposite sides are congruent, i.e., $\begin{align*}AB = CD\end{align*}$ and $\begin{align*}BC = DA\end{align*}$.

3. All the four angles are congruent and measure $\begin{align*}90^\circ\end{align*}$, i.e., $\begin{align*}\angle ABC\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle{BCD}\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle{CDA}\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle{DAB}\end{align*}$$\begin{align*}= 90^\circ\end{align*}$.

4. Diagonals are congruent, i.e., $\begin{align*}AC = BD\end{align*}$.

5. Diagonals bisect each other, i.e., $\begin{align*}BO = OD\end{align*}$ and $\begin{align*}AO = OC\end{align*}$.

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

 

The Properties of a Rhombus:

1. Opposite sides are parallel, i.e., $\begin{align*}AB \parallel CD\end{align*}$ and $\begin{align*}BC \parallel DA\end{align*}$.

2. All four sides are congruent, i.e., $\begin{align*}AB = BC = CD = DA\end{align*}$.

3. Opposite angles are congruent, i.e., $\begin{align*}\angle{ABC} = \angle{CDA}\end{align*}$ and $\begin{align*}\angle{DAB} = \angle{BCD}\end{align*}$.

4. Diagonals are the interior angle bisectors, i.e., $\begin{align*}\angle BAC = \angle DAC\end{align*}$, $\begin{align*}\angle BDC = \angle BDA\end{align*}$, $\begin{align*}\angle{DBC} = \angle{DBA}\end{align*}$ and $\begin{align*}\angle BCA = \angle DCA\end{align*}$.

5. Diagonals intersect each other at right angles, i.e., $\begin{align*}\angle AOB\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle BOC\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle COD\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle DOA\end{align*}$$\begin{align*}= 90^\circ\end{align*}$.

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

 

The Properties of a Square:

1. All four sides are congruent, i.e., $\begin{align*}AB = BC = CD = DA\end{align*}$.

2. All the four angles are congruent and measures $\begin{align*}90^\circ\end{align*}$, i.e., $\begin{align*}\angle ABC\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle BCD\end{align*}$$\begin{align*} = \end{align*}$$\begin{align*}\angle CDA\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle DAB\end{align*}$$\begin{align*}= 90^\circ\end{align*}$.

3. Diagonals are congruent, i.e., $\begin{align*}AC = BD\end{align*}$.

4. Diagonals bisect each other at right angles, i.e., $\begin{align*}\angle AOB\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle BOC\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle COD\end{align*}$$\begin{align*}=\end{align*}$$\begin{align*}\angle DOA\end{align*}$$\begin{align*}= 90^\circ\end{align*}$.

All the angles of a quadrilateral are congruent and make right angles. What type of quadrilateral is this?

All the angles of a _______ (kite/trapezoid/rectangle) are congruent and make right angles.

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.

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

Example 1

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

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. In other lessons, you will learn how these conjectures may be proven true.

Here are some possible conjectures:

1. diagonals of a square are congruent

2. diagonals of a square are perpendicular

3. diagonals of a square bisect each other (cut each other in half)

4. 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.

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