Understanding quadrilaterals

UNDERSTANDING QUADRILATERALS

Definition

A plane figure bounded by four line segments AB,BC,CD and DA is called a quadrilateral.

A B

D C

In geometry, a quadrilateral is a polygon with four sides and four vertices. Sometimes, the term

quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistence

with pentagon.

There are over 9,000,000 quadrilaterals. Quadrilaterals are either simple (not self-

intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave.

Parallelogram

Trapezium

Kite

Types of Quadrilaterals

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These figures are not polygons These figures are polygons

Definition: A closed figure formed by line segments so that each segment intersects exactly two others, but only at their endpoints.

Polygons

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Classifications of a Polygon

Convex: No line containing a side of the polygon contains a point in its interior

Concave:

A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

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Regular: A convex polygon in which all interior angles have the same measure and all sides are the same length

Irregular: Two sides (or two interior angles) are not congruent.

Classifications of a Polygon

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Polygon Names

3 sides Triangle

4 sides

5 sides

6 sides

7 sides

8 sides

Nonagon

Octagon

Heptagon

Hexagon

Pentagon

Quadrilateral

10 sides

9 sides

12 sides

Decagon

Dodecagonn sides n-gon

The formula we use to find the sum of the interior angles of any polygon comes from the number of triangles in a figure

First remember that the sum of the interior angles of a polygon is given by the formula 180(n-2).A polygon is called a REGULAR when all the sides are congruent and all the angles are congruent.The picture shown to the left is that of a Regular Pentagon. We know that to find the sum of its interior angles we substitute n = 5 in the formula and get:180(5 -2) = 180(3) = 540°

Regular triangles - EquilateralAll sides are the same length (congruent) and all interior angles are the same size (congruent).To find the measure of the interior angles, we know that the sum of all the angles equal 180°, and there are three angles.So, the measure of the interior angles of an equilateral triangle is 60°.

Quadrilaterals – squaresAll sides are the same length (congruent) and all interior angles are the same size (congruent)To find the measure of the interior angles, we know that the sum of the angles equal 360°, and there are four angles, so the measure of the interior angles are 90°.

Pentagon – a 5-sided polygonTo find the sum of the interior angles of a pentagon, we divide the pentagon into triangles. There are three triangles and because the sum of each triangle is 180° we get 540°, so the measure of the interior angles of a regular pentagon is 540°

Octagon – an 8-sided polygonAll sides are the same length (congruent) and all interior angles are the same size (congruent).What is the sum of the angles in a regular octagon?

Nonagon – a 9-sided polygonAll sides are the same length (congruent) and all interior angles are the same size (congruent).

What is the sum of the interior angles of a regular nonagon?

Decagon – a 10-sided polygonAll sides are the same length (congruent) and all interior angles are the same size (congruent).

What is the sum of the interior angles of a regular decagon?

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polygons

Classification by Sides with Flow Charts & Venn Diagrams

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral

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polygons

Classification by Angles with Flow Charts & Venn Diagrams

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangular

Obtuse

obtuse

• 2 sets of parallel sides• 2 sets of equal sides• Opposite angles equal• Adjacent angles supplementary• Diagonals bisect each other• Diagonals form 2 congruent triangles

Parallelogram

Types of Parallelograms

*RectangleI have all of the properties of the

parallelogram PLUS- 4 right angles

- diagonals congruent

*RhombusI have all of the properties of the

parallelogram PLUS- 4 congruent sides

- diagonals bisect angles- diagonals perpendicular

*SquareHey, look at me!

I have all of the properties of the parallelogram AND the rectangle

AND the rhombus.I have it all!

Is a square a rectangle?

• Some people define categories exclusively, so that a rectangle is a quadrilateral with four right

angles that is not a square. • This is appropriate for everyday use of the words, as people typically use the less specific word only when the more specific word will not

do. • Generally a rectangle which isn't a square is an

oblong.

• But in mathematics, it is important to define categories inclusively, so that a square is a

rectangle. • Inclusive categories make statements of

theorems shorter, by eliminating the need for tedious listing of cases.

• For example, the visual proof that vector addition is commutative is known as the

"parallelogram diagram". • If categories were exclusive it would have to

be known as the "parallelogram (or rectangle or rhombus or square) diagram"!

Trapezium

I have only one set of parallel sides. [The median of a trapezium is parallel to the bases and equal to one-half the sum of the

bases.]

Trapezoid Regular Trapezoid

• It has two pairs of sides.Each pair is made up of adjacent sides (the sides meet) 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.

Kite

Cyclic quadrilateral: the four vertices lie on a circumscribed circle. Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible. Bicentric quadrilateral: both cyclic and tangential.                                                                                                    

Some other types of quadrilaterals

Angle Sum Property Of Quadrilateral

The sum of all four angles of a quadrilateral is

360.. A

B C

D

1

23 4

6

5

Given: ABCD is a quadrilateral

To Prove: Angle (A+B+C+D) =360.

Construction: Join diagonal BD

Thank you for your attention