Quadrilaterals

A quadrilateral is a closed plane figure bounded by four line segments. For example, the figure ABCD shown here is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral. For example, AC is a diagonal of quadrilateral ABCD, and so is BD.

Naming Quadrilaterals

quadrilateral ABDC-wrong

quadrilateral ABCD-correct

In naming quadrilaterals, its vertices shoul be in consecutive order

Main Parts:

(based from the figure above)

Vertices:

Points: A, B, C, D

Segments:

segment AB

segment BC

segment CD

segment DA

Angles:(can be named by single letter)

angle A / DAB

angle B / ABC

angle C / BCD

angle D / CDA

Types of Quadrilaterals

There are special types of quadrilateral:

Properties of Quadrilateral

• Four sides (edges)
• Four vertices (corners)
• The interior angles add up to 360 degrees:

The Rectangle 

		means "right angle"
	and	show equal sides

A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

The Rhombus

A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines in second figure) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.

The Square

		means "right angle"
		show equal sides

A square has equal sides and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Parallelogram

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

A parallelogram with: all sides equal and angles "a" and "b" as right angles is a square!

The Trapezoid (UK: Trapezium)

Trapezoid	Isosceles Trapezoid

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.

It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.

And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:

	Trapezoid	Trapezium
US:	a pair of parallel sides	NO parallel sides
UK:	NO parallel sides	a pair of parallel sides

The Kite

Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.Sum of the Interior Angles

Sum of the Interior Angles

Prove that the angle sum of a quadrilateral is equal to 360º.

Proof:

Hence the angle sum of a quadrilateral is 360º.

Applying Properties of Angles in Quadrilaterals

The theorems we have proved can be used to prove other theorems. They can also be used to find the values of the pronumerals in a problem.

Problem 1

Find the value of the pronumeral x in the accompanying diagram. Give reasons for your answer.

Solution:

Problem 2

Find the value of each of the pronumerals in the kite shown here. Give reasons for your answers.

Solution:

Problem 3

Find the value of each of the pronumerals in the accompanying diagram. Give reasons for your answers.

Solution: