Quadrilateral
Definition: Let A, B, C and D are four points in a plane such that:
- No three points should be collinear.
- The line segments AB, BC, CD, and DA do not intersect except at their ends points
We can find area of quadrilateral ,Then the figure made up of the four lines segments is called the quadrilateral with vertices A, B, C, and D.
In the above figure:
Sides: AB, BC, CD, and DA are the sides.
Adjacent sides: (AB, BC), (BC, CD), (CD, DA), (DA, AB) are the four pairs of adjacent sides.
Opposite sides: (AB, CD) and (AD, BC) are the two pairs of the opposite sides.
Diagonals: AC and BD are two diagonals.
Angles: LABC , LBCD ,LCDA, LDAB are four angles. In which ( LA andLC) , (LB and LD) are opposite angles.
Polygon
A polygon is a plane shape with straight sides. the following figures shows a regular polygon,
Types of Polygons
1. Simple or Complex
A simple polygon has only one boundary and it does not cross over itself. A complex polygon intersects itself.The following figures a simple and complex polygon.
2. Concave and Convex Polygon
A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave.
3. Regular and Irregular Polygon
If all angles are equal and all sides are equal then it is regular polygon, otherwise it is irregular polygon.
The General Rule of Finding Interior Angles of a Regular Polygon.
| Name of the polygon | Number of sides | Measurement of Interior Angles |
| Triangle | 3 | 60o |
| Quadrilateral | 4 | 90o |
| Pentagon | 5 | 108o |
| Hexagon | 6 | 120o |
| Heptagon | 7 | .... |
| Octagon | 8 | .... |
| Nonagon | 9 | .... |
| Decagon | 10 | .... |
| N-gon | n | (n-2)180o /n |
Note: Each exterior angle of a regular polygon of n sides is equal to $\frac{360°}{n}$
Sum of Interior angles of a regular Polygon.
| Name of polygon | Number of Sides | Sum of Interior Angles |
| Triangle | 3 | 180o |
| Quadrilateral | 4 | 360o |
| Pentagon | 5 | 540o |
| Hexahon | 6 | 720o |
| Heptagon | 7 | .... |
| Octagon | 8 | (n-2)×180o |
Rule of Finding Diagonal of a Convex Polygon
An n-sided convex polygon has $\frac{n(n-3)}{2}$ diagonals.
A quadrilateral has $\frac{4(4-3)}{2}$ = 2 diagonals.
A regular hexagon has $\frac{6(6-3)}{2}$ = 9 diagonals. A triangle has no diagonal.
Some Example on Interior and Exterior Angles of a Regular Polygon.
Example 1. Find the value of x in the adjacent figure.
Solution: we know that the sum of the measures of exterior angles of a polygon is 360o
x + 90o + 50o + 110o = 360o
= x + 250o = 360o
= x = 360o – 250o
So, x = 110o
Example 2. Find the number of sides of a regular polygon whose each exterior angle has measure 45o
Solution: We know that the measure of each exterior angle of n-sided regular polygon is $\frac{360°}{n}$
$\frac{360°}{n}$ = 45
So, n = $\frac{360°}{45}$ = 8
Hence, there are 8 sides of the polygon.
Example 3. The interior angle of a regular polygon is 156o. Find the number of sides of the polygon.
Solution: Let there be n side of the polygon. Then, its each interior angle is equal to $\frac{180(n-2)}{n}$
$\frac{180(n-2)}{n}$ = 156
180n—360 =156n
24n = 360
n = 15
Thus, there are 15 sides of the polygon.
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