Polygon
A polygon is a closed figure formed by three or more than three line segments. These line segments are called the sides of the polygon. Each side of the polygon intersects exactly two other sides at their respective endpoints. The points of intersection of the sides of the polygon are known as vertices of the polygon. The term "polygon" is used to refer to a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180.

• Convex polygon: A type of polygon in which none of the interior angles is more than 180, is called a convex polygon.
• Concave polygon: A type of polygon in which at least one of the interior angles of the polygon is more than 180, is called concave polygon. 
• Regular polygon: A type of polygon which has all its angles and sides equal is called a regular polygon.

Classification of Polygons
Depending on the  number of sides polygons may be classified as follows:

• Triangle: A triangle is a three-sided polygon. Depending on the length of sides and measures on angles triangles may be equivalent, isosceles, scalene, acute triangle, obtuse triangle and right-angled triangle.
• Quadrilateral: The Four-sided polygons are called quadrilaterals. Can you guess the names of commonly known quadrilaterals ?
• Pentagon: The Five-sided polygons are called pentagons.
• Hexagon: The Six-sided polygons are called Hexagons
• Heptagon: The Seven-sided polygons are called Heptagons.
• Octagon: The Eight-sided polygons are called Octagons.
• Nonagon: The Nine-sided polygons are called Nonagons.
• Decagon: The Ten-sided polygons are called Decagons.

Polyhedrons
The shapes we studied above are two-dimensional shapes. Each of these shapes is measured in terms of length and breadth. These shapes are planar in dimensions and miss the height attribute. Now we shall study the three-dimensional counterparts of the shapes studied above. Three-dimensional shapes are those objects which have all the three-dimensions that is length, breadth, and height. These objects are also called solid shapes .

Polyhedrons are three-dimensional polygons. A polyhedron in simpler words is a solid-shaped polygon. An edge is that segment where each of the faces meets while vertices are those points where these edges meet. Two most common polyhedrons are pyramids and prisms .

Pyramids have triangular faces meeting at the vertex while prisms have rectangular faces as part of their lateral surfaces with top and bottom identical polygons.

Can you sight some more commonly sighted polyhedrons? Let us help you with it! A matchbox or an ice-cream cone is an example of a polyhedron.

Types of Angles
1. Acute Angle
An angle which is less than 90 ° is known as an acute angle. Acute angle measures between 0 to 90 °. The following figure shows that when two sides A and B intersect at point O and form a vertex. The angle at which these rays intersects is less than 90° thus forms an acute angle.

2. Obtuse Angle
An angle which is greater than 90 ° is known as an obtuse angle. An obtuse angle measure between 90 ° to 180 °. An obtuse angle is an opposite angle of an acute angle. The following figure shows that point O where A and B intersects is greater than 90° and less than 180°, thus it is an obtuse angle.

3. Right Angle
The right angle is an angle which measures exactly 90 °and any angle less or greater than 90 °will not be considered as a right angle. The following figure illustrates that A and B intersect to form a 90 ° angle.

4. Straight Angle
An angle which measures 180 ° is known as a straight angle. This looks similar to a plane straight line and hence is named as a straight angle. The following figure will clear the concept.

5. Reflex Angle
An angle which measures greater than 180 degrees but less than 360 degrees is known as a reflex angle. A reflex angle is supposed to be a complementary angle for acute angle and is on the other side of the line. The following figure illustrates the reflex angle.

Geometry finds its basis in angles. From basic closed shapes to tricky trigonometry questions, angles form a part of each and every chapter. Understanding these surely helps in perfecting knowledge of geometry and trigonometry both.

Triangle
Triangle is a three-sided polygon. This closed figure consists of three lines which are linked at the endpoint to each other. The main properties of the triangle are that the sum of the interior angle of the triangle will be 180° and the sum of the exterior angles will be 360° always. Triangle can be classified according to their sides and interior angles. Following are the details about each qualification.

Types of Triangles with Respect to Sides
1) Equilateral Triangle
This type of triangle consists of three equal sides and equal angles. Every side of the triangle is of the same length and every angle will be of the same measure of 60°. The following figure is an equilateral triangle –

2) Isosceles triangle
The triangle with only two equal sides is known as the isosceles triangle. Not only two equal sides, the isosceles triangle also consists of two equal angles. The following figure is of the isosceles triangle-

3) Scalene Triangle
The triangle with no equal sides is the scalene triangle. Each line of this triangle is of different length. Following is the figure of the scalene triangle:

Types of Triangles with Respect to Angles
1) Right Triangle
This triangle consists of one right angle and two acute angles. The right angle is an angle which measures 90 degrees and the acute angles are those angles which measure less than 90 degrees. Thus Right triangle is a triangle with one angle of 90° and the other two angle that measures less than 90°are acute angles. The right triangle is an angle with one 90° angle and two less than 90° angles. The following figure shows the right triangle-

2) Obtuse Triangle
This is a triangle with two acute angles and one obtuse angle. The acute angle is those angles which measure less than 90° whereas the obtuse angle is those angle which measures above 90°. Therefore, an obtuse triangle is a combination of two acute angles and one obtuse angle.

3) Acute Triangle
An acute triangle is a triangle which consists of three acute angles. This is the triangle which consists of three angles all of below 90

Quadrilaterals
Quad means four and lateral means sides. All closed figures with four sides are called quadrilaterals. The sides of the quadrilateral may be equal, unequal, parallel or irregular which forms the basis of varying shapes in these four-sided shapes. Whatever may be the external shape, every quadrilateral has four sides, four vertices and with all its angles adding up to be 360 °.

The most common types of quadrilaterals are Squares and Rectangles. We have been studying about these basic quadrilaterals since our primary classes, but in this part of the chapter we shall bring in light various other types of the quadrilateral, which show properties like square or rectangle yet are not named so.

Types of Quadrilaterals
The various types of quadrilaterals are
1. Parallelogram
As the name suggests, a parallelogram is a quadrilateral with parallel opposite sides. In a parallelogram all the opposite sides are equal and the angles made by each opposite side is also equal. In the figure shown below, PQRS is a quadrilateral and since PQ || RS and PS || QR we call it a parallelogram

2. Rectangle
The Rectangle is also a type of parallelogram with a little difference in the angles of the sides. In a rectangle, all the sides are at the right angle or to say perpendicular to each other. The figure below shows a quadrilateral ABCD and since all its sides are parallel to each other we may also call it a type of parallelogram with line segments placed at right angle to each other. This can be written as AB || CD and AC || BD. ∠A =∠B =∠C=∠D= 90°

3. Square
A square is a quadrilateral with all four sides equal. Every vertex of a square makes a right angle with its sides. A square is a quadrilateral which shows the following characteristic:
WX || YZ and WY || XZ. ∠W =∠X =∠Y=∠Z= 90°

4. Rhombus
A rhombus is also a parallelogram but with all sides equal. In other words, a parallelogram that shows  resemblance to a square as regards its lengths of sides is called a rhombus. These are sometimes referred to as Diamonds also. Rhombus shows a very peculiar feature regarding their diagonals. The diagonals in a rhombus meet exactly in the middle at a right angle and are said to bisect with each other.

In the figure above we see that STUV is a quadrilateral with sides ST || UV and SV || TU. All its sides are equal, thus ST=TU=UV=VS which also happen to form equal angles.

5. Trapezium
Amongst the other types of quadrilaterals, a trapezium shows different characteristics, In a trapezium, only one pair of the four sides are parallel to each other. In a trapezium, all the sides may not be equal in measurement but one of the pairs of sides has to be parallel to each other.

In the above figure, we can see a trapezium with sides AB || DC. A trapezium does not show similarity in lengths and angles of sides hence is an irregular quadrilateral.

Properties of Quadrilaterals
A four-sided polygon along with two dimensions is known as Quadrilateral. A Quadrilateral can be convex or conclave based on their dimensions. If Dimensions are inside the borderline of polygon than it will be termed as a convex quadrilateral. Based on the sides, and angle there are the following six types of convex Quadrilaterals.

• Parallelogram
• Rectangle
• Square
• Rhombus
• Trapezium
• Kite

But before elaborating the properties of the above quadrilateral we will go through a few common properties of quadrilateral that will be visible in every type of quadrilaterals-

• Every quadrilateral has four sides.
• Every quadrilateral consist of four corners or vertex
• The interior angle of quadrilateral adds up to 360° in every quadrilateral.

1) Properties of Parallelogram

• Opposite sides are equal and parallel to each other.
• Opposite angles are equal.
• The adjacent angles are supplementary.
• Diagonals bisect each other.
  

2) Properties of Rectangle

• Opposite sides are equal.
• All angles in the rectangle are the right angle.
• Diagonals are congruent and bisect each other.
  

3) Properties of Square

• All sides are equal in square.
• Every interior angle will be of 90°.
• Diagonals are perpendicular and bisect each other.

4) Properties of Rhombus

• All sides are congruent.
• Diagonals intersect each other and are perpendicular to each other.
• Opposite angles are congruent.
• Adjacent angles are supplementary.
  

5) Properties of Kite

• Two pairs of adjacent size are congruent.
• The angle between unequal sides are equal
• Diagonals intersect each other.
• One diagonal is perpendicular to another.

6) Properties of Trapezium

• One pair of opposite side is parallel.
• Diagonals intersect each other in the same ratio.
• Two adjacent angles are supplementary.

After going through the above properties of every quadrilateral it will now be easier for you to differentiate between them. So every quadrilateral is a four-sided figure but still, there is a lot of difference between them depending on the length of sides and the equality of angle. The difference between them is mentioned in their properties, so go through the above properties carefully as you are now able to differentiate between every type of quadrilateral.

Some examples solutions
Example 1:  In the adjoining figure, PQRS is a parallelogram. Find x and y in cm

Solution: In a parallelogram, we know that the diagonals bisect each other. Therefore,
SO = OQ
This given,16 = x + y
Similarly, PO = OR,
So that 20 = y + 7
We obtain y = 20 – 7 = 13 cm.
Substituting the value of y in the first relation, we get 16 = x + 13. Hence x = 3 cm.

Example 2: The figure shown below is of a straight angle, ∠AOB. Find the missing angle in the given figure:

Solution: ∠AOB is a straight angle, that is ∠AOB = 180°
∠AOB = ∠AOC +∠COB
180 = ?+55
? = 180-55
? = 125 °
The missing angle in the above figure: 125°

Example 3:  An equilateral triangle can also be called an acute angle, why ?
Solution: All the three sides and angles of an equilateral triangle are equal. We know that the sum of angles in a triangle is 180°. Since all the three angles in an equilateral triangle are equal, we write it as:
∠A + ∠B +∠C = 180
Now, ∠A = ∠B = ∠C, hence the sum of three angles can be written as ∠A + ∠A + ∠A = 3 ∠A
3 ∠A = 180°
∠A = 180/3
= 60°
Since all the angles in an equilateral triangle are equal, we get, ∠A = ∠B = ∠C = 60°

Example 4: A bicycle tyre has 20 spokes, the angle between a pair of adjacent spokes is :
Solution: Option B. A circle is a polygon with infinite sides, but the total angle that forms a circle is 360°. Now as given in the question, the bicycle has 20 spokes, so the angle between each pair of adjacent spokes is 360 ⁄ 20 = 18°

Example 5: The perimeter of an isosceles right angled triangle is 40m, Find the area of this particular triangle.
Solution : In an isosceles right angled triangle, Height = Base. Let a be the base of this triangle and b be the hypotenuse of this triangle
a + a + b = 40 2a + b = 40.
Also b² = a² + a²
b² = 2a².
b = √a. So 2a + √2a = 40⇒ 3.141a = 40
a = 12.73m.
Required area = (1/2)a² = (1/2)×(12.73)(12.73)=81.02 m²