Polygons and quadrilaterals

Rhombus

A rhombus is a parallelogram that has all four sides of equal length. A summary of the properties of a rhombus is:

• Both pairs of opposite sides are parallel.
• All sides are equal in length.
• Both pairs of opposite angles are equal.
• Both diagonals bisect each other at ${90}^{\circ }$ .
• Diagonals of a rhombus bisect both pairs of opposite angles.

Square

A square is a rhombus that has all four angles equal to 90 ${}^{\circ }$ .

A summary of the properties of a square is:

• Both pairs of opposite sides are parallel.
• All sides are equal in length.
• All angles are equal to ${90}^{\circ }$ .
• Both pairs of opposite angles are equal.
• Both diagonals bisect each other at ${90}^{\circ }$ .
• Diagonals are equal in length.
• Diagonals bisect both pairs of opposite angles (ie. all ${45}^{\circ }$ ).

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal.

A summary of the properties of a kite is:

• Two pairs of adjacent sides are equal in length.
• One pair of opposite angles are equal where the angles are between unequal sides.
• One diagonal bisects the other diagonal and one diagonal bisects one pair of opposite angles.
• Diagonals intersect at right-angles.

Rectangles are a special case (or a subset) of parallelograms. Rectangles are parallelograms that have all angles equal to 90. Squares are a special case (or subset) of rectangles. Squares are rectangles that have all sides equal in length. So all squares are parallelograms and rectangles. So if you are asked to prove that a quadrilateral is a parallelogram, it is enough to show that both pairs of opposite sides are parallel. But if you are asked to prove that a quadrilateral is a square, then you must also show that the angles are all right angles and the sides are equal in length.

Polygons

Polygons are all around us. A stop sign is in the shape of an octagon, an eight-sided polygon. The honeycomb of a beehive consists of hexagonal cells. The top of a desk is a rectangle. Note that although in the first two of these cases the sides of the polygon are all the same, but this need not be the case. Polygons with all sides and angles the same are called 'regular', while those with some sides or angles that are different are called 'irregular'. Although we often work with irregular triangles and quadrilaterals, once one gets up to polygons with greater than four sides, the more interesting ones are often the regular ones.

In this section, you will learn about similar polygons.

Similarity of polygons

Discussion : similar triangles

Fill in the table using the diagram and then answer the questions that follow.

1. What can you say about the numbers you calculated for: $\frac{\mathrm{AB}}{\mathrm{DE}}$ , $\frac{\mathrm{BC}}{\mathrm{EF}}$ , $\frac{\mathrm{AC}}{\mathrm{DF}}$ ?
2. What can you say about $\widehat{A}$ and $\widehat{D}$ ?
3. What can you say about $\widehat{B}$ and $\widehat{E}$ ?
4. What can you say about $\widehat{C}$ and $\widehat{F}$ ?

If two polygons are similar , one is an enlargement of the other. This means that the two polygons will have the same angles and their sides will be in the same proportion.