TRIANGLE PROPERTY SOLUTIONS

A Triangle is basically a polygonal structure that has three edges, three angles, and three respective vertices.

On the basis of sides, we can classify triangles as:

1. Equilateral Triangle.
2. Scalene Triangle
3. Isosceles Triangle.

On the basis of angles, we can classify triangles into:

1. Acute-angled Triangle.
2. Obtuse-angled Triangle.
3. Right-angled Triangle.

What are the Properties of a Triangle?

Properties of Triangle is an essential item in the study of Geometry. It is crucial to have a full-fledged understanding of its properties so that it can be effectively applied while dealing with practical problems.

Properties of Triangles:

1.  The sum of all the three angles of a triangle is equal to 180º.

In a triangle ABC, 

∠A +∠B +∠C =180º.

2.As per the theorem of triangle inequality the sum of the length of any two sides has to be greater in a triangle.

In a triangle ABC, 

         a+b>c,

          b+c>a,

          c+a>b.

3.The side which lies right opposite to the greater angle will be the longest side of the triangle.

In a triangle ABC, 

  If ∠B is the greatest angle, the side opposite to it i.e., AC will be the longest side.

4.The Pythagoras property or what is commonly known as the Pythagoras theorem states that the square of the hypotenuse will be equal to the sum of the square of the two other sides in a right-angle triangle.

   Base² + Height²= Hypotenuse².

5.The congruence property of a triangle states that all the corresponding sides, as well as all the angles, must be equal for a triangle to be congruent.

If we take two triangles, say ABC and XYZ, then, 

             ∠ABC=∠XYZ.

             ∠BAC=∠YXZ.

              ∠CAB=∠ZXY.

               AB=XY.

                AC=XZ.

                 BC=YZ.

6.The exterior angle of a triangle is the sum total of the angles of the two opposite interior angles.

If the exterior angle is e and the interior angles are a,b,c, then

    ∠e=∠a+∠b.

Now, we will discuss the properties of types of triangles:

Equilateral Triangle:

1. Identical length on all the three sides
2. The angle at each end will also be equal.
3. The Interior angle of an equilateral triangle will be 60ºeach.

  Scalene Triangle:

1. There are no equal sides and angles.
2. It contains one obtuse angle.

 Isosceles Triangle:

 Has two sides of equal length and the other side is of different length.

  Acute-angled Triangle:

1. Each angle of the triangle will be less than 90º.
2. Unequal sides.

Obtuse-angled Triangle:

1. One angle will be more than 90º.
2. The other two angles will be less than 90º.

Right-angled Triangle:

1. One particular angle will be 90º.
2. The largest side will be the opposite of 90ºalso known as hypotenuse.

Few Solved Examples:

Property 1: 

Example:

Find the value of x if (x-10), xº and (x+40)ºrepresent three interior angles of a triangle.

Solution:

We know that, 

Sum of all the three angles = 180º.

Let the sides of the triangle XYZ be XY=(x-10)º, YZ=xºand ZX=(x+40)º.

According to the problem, 

               (x-10)º+ xº+ (x+40)º=180º.

             = 3x+30º=180º.

              =3x=180º-30º.

              =3x= 150º.

                =x=150º/3=50º.

Thus, the value of x= 50º.

Putting the value of x in,

         (x-10)º=50º-10º=40º.

          (X+40)º=50º+40º=90º.

Thus, 50º+40º+90º=180º.

Property 2: Triangle Inequality

Example:

3 units, 4 units, and 5 units are the respective lengths of three sides of a triangle. Prove that it satisfies the triangle inequality property.

Solution:

Let ABC be a triangle, where

AB= 5 units, BC=3 units, and CA=4 units.

According to the theorem, the length of any two sides should be greater than the third side.

Therefore, AB+CA>BC

                   = 5 units+ 4units> 3 units.

                     AB+ BC > CA

                    = 5 units+ 3 units> 4 units.

                      BC+CA>AB

                      = 3 units+ 4 units > 5 units .

Thus, the above example satisfies the triangle inequality theorem.

Property 3: Pythagoras Theorem

Example:

The length of the sides of a triangle are 5cm, 4cm, and xcm. Find x.

Solution

According to pythagoras theorem, 

 Hypotenuse²=Base²+Height².

Let us assume ABC as a triangle, where, AB= 5cm, BC= xcm, and CA=4 xcm.

Therefore, (CA)²=(AB)²+ (BC)².

                   = 4²=5²+x².

                 = x²= 25-16

             =x= √9

               = x= 3 cm.

Conclusion

The properties of triangles help us determine unknown quantities in geometry and trigonometry.

They are usually applied in fields such as construction, navigation astronomy, and architecture.

They cover an entire gamut of geometrical properties and help us solve complex problems in various fields.