Here are the facts about quadrilaterals that you need to know.
The four angles in a quadrilateral add up to 360 degrees.
You can use this property to solve for unknown angles.
Example In the diagram below,
To find \( \angle \theta \),
\[ \begin{align} \angle ABC &= 75^o \quad \text{ (alternate angles AB||DC)} \\ \angle BCD &= 105^o \quad \text{ angles in a straight line} \\ \theta &= 360^o - 80^o - \angle ABC - \angle BCD \\ &= 100^o \end{align}\]
A Parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Properties of Parallelograms
If you are told that a figure is a parallelogram you are entitled to use the following facts:
Parallelograms have three special cases.
Rhombus
A rhombus is a parrallelogram with both pairs of opposite sides equal. It has the additional properties:
Conversely, if a parallelogram has any one of properties 1 and 2 then it is a rhombus.
Rectangle
A rectangle is a parallelogram with one right angle. It has the additional properties:
Conversely, if a parallelogram has any one of properties 1 and 2 then it is a rectangle.
Square
A square is a rectangle with one pair of adjacent sides equal. It has all the properties of a rectangle plus:
Conversely, if a rectangle has this property then it is a square.
Guided Examples
Example In the diagram below,
To find angle y,
\[ \begin{align} \angle y &= 2x + 30 \quad \text{ (opposite angles in a parallelogram)} \\ 2x + 30 + x &= 180 \quad \text{(co-interior angles, AD||BC)} \\ 3x + 30 &= 180 \\ 3x &= 150 \\ x &= 50 \\ y &= 2x+30 \\ &= 130^o \end{align}\] Note - there are many ways to solve this. This solution used properties of a parallelogram.
If you are given a quadrilateral then if it has the following properties you are entitled to conclude it is one of the following:
Parallelogram
Rhombus
Rectangle
Square
Example In the diagram below,
To find angle x,
ABCD is a parallelogram because diagonals bisect, so
\[ \begin{align} \angle CAD &= \angle BCA = 50^o \quad \text{(alternate angles, AD||BC)} \\ \angle AED &= 100^o \quad \text{(vert opposite angles)} \\ x &= 180 - 150 \quad \text{(angle sum of a triangle)} \\ &= 30^o \end{align}\]