Definition and general properties
Definition and vocabulary
Quadrilateral
A quadrilateral is a plane figure with four sides.
The following graphic contains several examples of quadrilaterals.
Diagonal
A line segment joining opposite angles of a quadrilateral is called a diagonal of the quadrilateral.
The following graphic contains several quadrilaterals and their diagonals.
General properties
Every quadrilateral has two diagonals.
Every quadrilateral has four angles. The sum of the four angles of a quadrilateral is 360^\circ.
Consider the following graphic which contains a quadrilateral and the measurements of three of its four angles:
We can solve for the missing angle x :
80^\circ+120^\circ+130^\circ+x=360^\circ\\x=360^\circ-330^\circ\\x=30^\circ
Special quadrilaterals
Parallelograms
Parallelogram
A parallelogram is a quadrilateral whose opposite sides are parallel to one another.
Perimeter of a parallelogram
Consider the following parallelogram:
The perimeter of the parallelogram is:
P=2\left(x+y\right)
Consider the following parallelogram:
The perimeter of the parallelogram is:
2\left(5+8\right)=2\left(13\right)=26
Area of a parallelogram
Consider the following parallelogram:
The area of the parallelogram is:
A=h\ell
Consider the following parallelogram:
The area of the parallelogram is:
9\left(7\right)=63
Second area formula of a parallelogram
Consider the following parallelogram:
The area of the parallelogram is:
A=xy\sin\left(\alpha\right)
Consider the following parallelogram:
The area of the parallelogram is:
7\left(8\right)\sin\left(150^\circ\right)=56\left(\dfrac{1}{2}\right)=28
Any pair of opposite sides of a parallelogram have equal length.
Consider the following parallelogram:
The value of x is:
x=5
The value of y is:
y=5
Any pair of angles opposite to one another in a parallelogram have the same measure.
Consider the following parallelogram:
The measure of \alpha is:
\alpha=60^\circ
The measure of \beta is:
\beta=120^\circ
The sum of two adjacent angles in a parallelogram is 180^\circ.
Consider the following parallelogram:
Since the angle \alpha is adjacent to an angle of measure 30^\circ, we must have:
\alpha+30^\circ=180^\circ\\\alpha=150^\circ
Then by the previous theorem we have:
- \beta=30^\circ
- \gamma=\alpha=150^\circ
The two diagonals of a parallelogram bisect each other.
Consider the following parallelogram:
Since the diagonals of a parallelogram bisect each other, we can solve for x and y :
- x=8
- y=7
Rhombuses
Rhombus
A rhombus is a parallelogram whose sides all have equal length.
The following graphic contains several rhombuses.
A rhombus is a special type of parallelogram. Therefore, every theorem on parallelograms applies to rhombuses.
Consider the following rhombus:
Since the figure is a rhombus, all sides are of equal length. Therefore:
x=y=z=7
Because a rhombus is a parallelogram we can solve for the missing angles:
- \alpha+130^\circ=180^\circ\\\alpha=50^\circ
- \beta=130^\circ
- \gamma=\alpha=50^\circ
A parallelogram is a rhombus if and only if the diagonals are perpendicular to one another.
The following graphic illustrates the diagonals of a rhombus intersecting at 90^\circ.
Rectangles and squares
Rectangle
A parallelogram whose angles all measure 90^\circ is called a rectangle.
The following graphic contains several rectangles.
A rectangle is a special type of parallelogram. Therefore, every theorem on parallelograms applies to rectangles.
Area of a rectangle
Consider the following rectangle:
The area of the rectangle is:
A=bh
Consider the rectangle in the following graphic:
The area of the rectangle is:
8\left(3\right)=24
Square
A rectangle whose sides all have equal lengths is called a square.
Area of a square
Consider the following square:
The area of the square is:
A=h^2
Consider the square in the following graphic:
The area of the square is:
3^2=9
Trapezoids
Trapezoid
A quadrilateral with at least two sides parallel to one another is called a trapezoid.
Area of a trapezoid
Consider the following trapezoid:
The area of the trapezoid is:
A=\dfrac{1}{2}\left(b_1+b_2\right)h
Consider the following trapezoid:
The area of the trapezoid is:
\dfrac{1}{2}\left(2+8\right)4=20
Kites
Kite
A quadrilateral with two pairs of adjacent sides of equal length is called a kite.
The two angles formed by the sides of different length are congruent.
Consider the following kite:
We can solve for the missing angles. By the theorem we know:
\alpha=100^\circ
The sum of the angles of a quadrilateral is 360^\circ. Therefore, we can solve for the remaining angle:
\beta+100^\circ+100^\circ+50^\circ=360^\circ\\\beta=110^\circ
The diagonals of a kite are perpendicular.
Main and cross diagonals of a kite
- The main diagonal of a kite is the diagonal which connects the vertices with different angles.
- The cross diagonal of a kite is the diagonal which connects the vertices with congruent angles.
The main diagonal of a kite bisects the cross diagonal of a kite.
Area of a kite
The area of a kite is half the product of the lengths of the diagonals. Consider the following kite:
The area of the kite is:
A=\dfrac{1}{2}xy
Consider the following kite:
The area of the kite is:
\dfrac{1}{2}\left(7\right)\left(11\right)=\dfrac{77}{2}
Proofs involving quadrilaterals
The theorems in this lesson can be used to prove new properties of quadrilaterals.
Consider the following graphic of a square:
We can prove that the area of the square is \dfrac{h^2}{2}.
By the Pythagorean Theorem, we know that:
x^2+x^2=h^2\\2x^2=h^2\\x^2=\dfrac{h^2}{2}
The area of the square is x^2. Therefore, we have proven the claim.
In the following graphic, we can observe the following properties:
- The quadrilateral with vertices A,C,D,F is a parallelogram
- B is the midpoint of the line segment \overline{AC}
- E is the midpoint of the line segment \overline{DF}
We can prove that the quadrilateral with vertices A,B,E,F is a parallelogram with half the area of the parallelogram with vertices A,C,D,F.
- \alpha=\angle CDF because the quadrilateral with vertices A,C,D,F is a parallelogram.
- \overline{AB}=\dfrac{\overline{AC}}{2}=\dfrac{\overline{DF}}{2}=\overline{BE} because the quadrilateral with vertices A,C,D,F is a parallelogram, B is the midpoint of the line segment \overline{AC}, and E is the midpoint of the line segment \overline{DF}.
Therefore, the line segment \overline{BE} is parallel to the line segment \overline{AF} and has the same length.
The quadrilateral with vertices A,B,E,F is a parallelogram.
The area of the parallelogram with vertices A,B,E,F is:
\overline{AB}\cdot \overline{AF}\sin\left(\alpha\right)=\dfrac{\overline{AC}}{2}\overline{AF}\sin\left(\alpha\right)
The value \dfrac{\overline{AC}}{2}\overline{AF}\sin\left(\alpha\right) is the area of the parallelogram with vertices A,C,D,F.