isosceles triangle | right | obtuse | acute isosceles triangle

Definition of An isosceles triangle(समद्विबाहु त्रिकोण):
A Triangle with two equal side is an Isosceles Triangle(if two side of triangle equal in length then two angles of triangle also equal), So now Defintion is like "A Triangle with two equal side and Two equal Angles" is an isosceles triangl.lenght a=b and angle α is equal(see in below picture/image)

isosceles triangle theorem
if angles, then sides: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If sides, then angles: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Area and Perimeter of isosceles Triangle
perimeter of isosceles triangle=2a+b
Area of isosceles Triangle=1/2*b*h
here h=height of isosceles Triangle,b=base of isosceles Triangle



Types Of isosceles Triangles
acute isosceles triangle:
An ACUTE isosceles(all angles are acute) triangle contains three acute angles (each angle of an acute triangle is less than 90 degrees).sum of the angles of a triangle is always 180 degrees. An acute triangle may be isosceles, equilateral, or scalene.

obtuse isosceles triangle:
Obtuse isosceles triangles have one angle that is greater than 90° degrees.(Obtuse triangles have one obtuse angle).sum of the angles of a triangle is always 180 degrees.An obtuse triangle may be scalene or isosceles.

right isosceles triangle:
The right isosceles triangle has one 90 degree angle and two acute (< 90 degree) angles.sum of the angles of a triangle is always 180 degrees.A right triangle may be scalene or isosceles.

Obtuse triangles | Obtuse Angle

Obtuse triangles have one angle that is greater than 90° degrees.(Obtuse triangles have one obtuse angle).
sum of the angles of an obtuse triangle is always 180 degrees.
An obtuse triangle will have one and only one obtuse angle. The other two angles are acute angles. 
The sum of the two angles other than the obtuse angle is less than 90º. 
An obtuse triangle may be scalene or isosceles.
Figure:Obtuse triangle
Obtuse Angle Defintion

An Obtuse Angle is more than 90° but less than 180°.

acute triangle | Acute Angle

An ACUTE (all angles are acute) triangle contains three acute angles (each angle of an acute triangle is less than 90 degrees).

Acute Angle Definition
The acute angle is the small angle which is less than 90° but more than zero degrees
sum of the angles of an acute triangle is always 180 degrees.
An acute triangle may be isosceles,scalene or equilateral.
The right triangle has one 90 degree angle and two acute (< 90 degree) angles 
An equilateral triangle is a specific type of acute triangle where the three angles have an equal measure of 180°/3 = 60°.
Figure:Acute Triangle

right triangle | Right Angle | Special right triangles

The right triangle has one 90 degree angle and two acute (< 90 degree) angles.
sum of the angles of a right triangle is always 180 degrees.
A right triangle may be scalene or isosceles.
Figure:Right Triangle
special right triangles
There are two "special" right triangles, the 45º- 45º- 90º triangle and the 30º- 60º- 90º triangle.  The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions.

Right Angle
A right angle is an internal angle which is equal to 90°.

complementary angles definition examples

Definition: Complementary angles(संपूरक कोण) are two angles that have a sum of 90° (Right angle example of Complementary angles).when sum of two angles 90°, they complement each other. In Right triangle one angle of 90° and two other angles sum of 90°, So Right angle Example of Complementary angles.Figure 1 example of complementary angles,because Sum of angle A and B equal to 90°.
A complementary angle is made up of two acute angles.
Figure 1: Complementary angles
Figure 2: Complementary angles
These two angles in Figure 2 are complementary, because 60 + 30 = 90.Complementary angles form a right angle (L shape) and have a sum of 90 degrees.

Determine Complement Angle
To determine the complement, subtract the given angle from 90.
Example 1: If one angle is 48 degree, find the second angle if the two angles are complementary to each other.
Solution: 90 - 48 = 42°,  The complement of 48° is 42°.
Example 2: If one angle is 15 degree, find the second angle if the two angles are complementary to each other.
Solution: 90 - 60 = 30° ,  The complement of 60° is 30°.
Example 3: If one angle is 30 degree, find the second angle if the two angles are complementary to each other.
Solution: 90 - 30 = 60°,  The complement of 30° is 60°.
Example 4: If one angle is 15 degree, find the second angle if the two angles are complementary to each other.
Solution: 90 - 15 = 50° ,  The complement of 15° is 75°.
Practice Problems
Example 1: The measure of angle 1 is 27 degree and angle 2 is 60 degree. Identify whether the angles are complementary to each other or not?
Example 2: Let first angle is 12 degrees, find the second angle if the two angles are complementary to each other?
Example 3: x and y are complementary angles. Given x = 62˚, find the value y?
Example 4: The measure of angle 1 is 15 degree and angle 2 is 160 degree. Identify whether the angles are complementary to each other or not?
Example 5: Let first angle is 30 degrees, find the second angle if the two angles are complementary to each other?

Example 6: x and y are complementary angles. Given w = 75˚, find the value z?
Example 7: The measure of angle 1 is 45 degree and angle 2 is 90 degree. Identify whether the angles are complementary to each other or not?
Example 8: Let first angle is 60 degrees, find the second angle if the two angles are complementary to each other?

Example 9: x and y are complementary angles. Given a = 90˚, find the value b?
Example 10: The measure of angle 1 is 120 degree and angle 2 is 60 degree. Identify whether the angles are complementary to each other or not?

supplementary angles | Definition | Examples

Angles are Supplementary Angle(अधिक कोण) if Sum Of Two Angles 180 degrees. These two angles (135° and 45°) are Supplementary Angle, because sum of 135°+45°=180°.
Supplementary Angles together make a straight line, But the angles don't have to be together.
When the two angles sum 180°, they "Supplement" each other.
When two adjacent angles form a straight line, they are supplementary.
If one angle is known, its supplementary angle can be found by subtracting the measure of its angle from 180°
Fig 1 Example of Supplementary Angles, because sum of angle a and b are 180°. 
Fig 1: Supplementary Angles
Determine supplement Angle
To determine the Supplement, subtract the given angle from 180.
Example 1: If one angle is 50 degree, find the second angle if the two angles are supplementary to each other.
Solution: 180 - 50 = 130°,  The Supplement of 50° is 130°.
Example 2: If one angle is 60 degree, find the second angle if the two angles are supplementary to each other.
Solution: 180 - 60 = 120° ,  The Supplement of 60° is 120°.
Example 3: If one angle is 30 degree, find the second angle if the two angles are v to each other.
Solution: 180 - 30 = 150°,  The Supplement of 30° is 150°.
Example 4: If one angle is 15 degree, find the second angle if the two angles are supplementary to each other.
Solution: 180 - 15 = 165° ,  The Supplement of 15° is 165°.
some other examples of supplementary angle pairs
20 degrees and 160 degrees
40 degrees and 140 degrees
60 degrees and 120 degrees
70 degrees and 110 degrees
90 degrees and 90 degrees

Practice Problems
Example 1: The measure of angle 1 is 27 degree and angle 2 is 150 degree. Identify whether the angles are supplementary to each other or not?
Example 2: Let first angle is 12 degrees, find the second angle if the two angles are supplementary to each other?
Example 3: x and y are supplementary angles. Given x = 45˚, find the value y?
Example 4: The measure of angle 1 is 27 degree and angle 2 is 150 degree. Identify whether the angles are supplementary to each other or not?

Example 5: Let first angle is 22 degrees, find the second angle if the two angles are supplementary to each other?
Example 6: The measure of angle 1 is 10 degree and angle 2 is 120 degree. Identify whether the angles are supplementary to each other or not?

Example 7: Let first angle is 120 degrees, find the second angle if the two angles are supplementary to each other?
Example 8: The measure of angle 1 is 15 degree and angle 2 is 145 degree. Identify whether the angles are supplementary to each other or not?

Example 9: Let first angle is 12 degrees, find the second angle if the two angles are supplementary to each other?
Example 10: The measure of angle 1 is 30 degree and angle 2 is 135 degree. Identify whether the angles are supplementary to each other or not?

Coterminal Angles | find coterminal angle

Two angles are coterminal angle if they are drawn in the standard position and both have their terminal sides in the same location, To find negative and positive coterminal Angles with a given angle, you can subtract and add 360° if the angle is given in degrees or 2π if the angle is given in radians.See below diagram of Coterminal Angles, in which given angle 60° and Coterminal Angles of 60° is (60°-360°) and (360°+60°), negtive and positive coterminal respectivly.
Angles that are coterminal have the same value for function like sin, cos, tan, 60°, 420° and -300° are coterminal, and so sin60°, sin420° and sin(-300°) all have the same value(√3/2).
Fig: Coterminal Angles 

Examples of Coterminal Angles
Q.1 Find a positive and a negative angle coterminal with a 65° angle?
Ans. 65°−360°=−295° and 65°+360°=425°, −295° angle and a 425° angle are coterminal with a 65° angle. 
Q.2 Find a positive and a negative angle coterminal with a 45° angle?
Ans. 45°−360°=−315° and 45°+360°=405°, −315° angle and a 405° angle are coterminal with a 45° angle.
Q.3 Find a positive and a negative angle coterminal with a 95° angle?
Ans. 95°−360°=−265° and 65°+360°=455°, −265° angle and a 455° angle are coterminal with a 95° angle. 
Q.4 Find a positive and a negative angle coterminal with a 15° angle?
Ans. 15°−360°=−345° and 15°+360°=475°, −345° angle and a 475° angle are coterminal with a 15° angle.
Q.5 Find a positive and a negative angle coterminal with a 90° angle?
Ans. 90°−360°=−270° and 90°+360°=450°, −270° angle and a 450° angle are coterminal with a 90° angle. 
Q.6 Find a positive and a negative angle coterminal with a 30° angle?
Ans. 30°−360°=−330° and 30°+360°=390°, −330° angle and a 390° angle are coterminal with a 30° angle.

incenter of a Right | Acute | Obtuse Triangle

Definition:Triangle’s three angle bisectors(A bisector cuts an angle into two equal parts) intersect at a point and that point Known as incenter of a triangle.

Properties of incentre:
Draw a circle inside triangle(incircle), the largest circle that will fit inside the triangle and touch all three sides and centre of circle is known as an incenter of a triangle.

Triangle's incenter is always inside the triangle, it will never go outside the triangle
Incenter is equidistant(distance is equal) from all the sides of the triangle. 
The radius of the incircle is the length of the perpendicular line drawn from the Incenter to any side 

Figure 1: incenter and incircle of a triangle
Incentre and Incircle of Acute Triangle:
Acute Triangle’s three angle bisectors(A bisector cuts an angle into two equal parts) intersect at a point and that point is incenter of an Acute triangle.The largest circle that will fit inside the Acute triangle and touch all three sides Known as Incircle of Acute Triangle.
Figure 2: incenter and incircle of an acute triangle

Incentre and Incircle of Obtuse Triangle:
Obtuse Triangle’s three angle bisectors(A bisector cuts an angle into two equal parts) intersect at a point and that point is incenter of an Obtuse triangle.The largest circle that will fit inside the Obtuse triangle and touch all three sides Known as Incircle of Obtuse Triangle.
Figure 3: incenter and incircle of an obtuse triangle

Incentre and Incircle of Right Triangle:
Right Triangle’s three angle bisectors(A bisector cuts an angle into two equal parts) intersect at a point and that point is incenter of a Right triangle.The largest circle that will fit inside the Right triangle and touch all three sides Known as Incircle of Right Triangle.
Figure 4: incenter and incircle of a right triangle
Description
The incenter is the point of intersection of the three angle bisectors. The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles. The incenter is the center of the circle inscribed in the triangle.

equilateral triangle | equiangular | acute triangle

Definition of Equilateral Triangle (समभुज त्रिकोण):
Equilateral Triangle is a Triangle in which all three sides are equal in lenght,equilateral triangle are also equiangular(angles are equal to each other, which is possible if all angles is equal to 60 degree).Now we can define equilateral triangle as "a triangle in which Angles and lenght of three side is equal", given picture/image of equilateral triangle below

Area of an equilateral triangle:
s is the length of one side of the triangle, and 
Area
 of equilateral triangle=
3
4
s
2

Height/altitude of equilateral triangle Formula=s/2 x √3
Perimeter of equilateral triangle Formula=3s

Types of equilateral triangle
acute equilateral triangle:
An ACUTE equilateral(all angles are acute) triangle contains three acute angles (each angle of an acute triangle is less than 90 degrees).sum of the angles of a triangle is always 180 degrees. An acute triangle may be isosceles, equilateral, or scalene.

equiangular triangle:
In an equiangular triangle, all the angles are equal—each angle equal to 60 degrees. An equiangular triangle is a kind of acute triangle, and equiangular triangle is always equilateral.

Is it possible to have a equilateral right triangle?
No(because in right triangle all angles are not equal)
Is it possible to have a equilateral obtuse triangle?
No(because in obtuse triangle all angles are not equal)

scalene triangle | acute | obtuse | right scalene triangle

what is a scalene triangle(विषमभुज त्रिभुज):
Simple definition of scalene triangle is: A Triangle in which No equal angles and no equal sides

As you can see in above triangle all sides with different lengths(AB,BC,CA not equal in lenghts) and Angles also not equal(68,71,41), so a triangle with no equal sides and Angles called scalene triangle(see above image/picture of scalene triangle).sum of the angles of a triangle is always 180 degrees.

Area of a scalene triangle:
s = (a + b + c) / 2
area of scalene triangle = sqrt(s * (s - a) * (s - b) * (s - c))

Types Of scalene Triangles:
acute scalene triangle:
An ACUTE scalene(all angles are acute) triangle contains three acute angles (each angle of an acute triangle is less than 90 degrees).sum of the angles of a triangle is always 180 degrees.An acute triangle may be isosceles,scalene or equilateral.

obtuse scalene triangle:
Obtuse scalene triangles have one angle that is greater than 90° degrees.(Obtuse triangles have one obtuse angle).sum of the angles of a triangle is always 180 degrees.An obtuse triangle may be scalene or isosceles.

right scalene triangle:
The right scalene triangle has one 90 degree angle and two acute (< 90 degree) angles.sum of the angles of a triangle is always 180 degrees.A right triangle may be scalene or isosceles.