Triangles

A triangle can be defined as a three-sided polygon with three inner angles. It is one of the most fundamental forms in geometry, consisting of three vertices linked together and symbolized by the symbol △. 

Types of Triangles

The sides and angles of a triangle are used to classify it into several parts. Triangles may be divided into several groups based on their sides and angles. A right triangle, for example, has one angle of 90 degrees, but an equilateral triangle has all sides of equal length.

• 3 sides of the equilateral triangle are the same length magnitude. with all angles equal to 60 degrees.
• Now, In an isosceles triangle, Two sides are of equal length. Also, it has two angles of equal length, namely the angles opposing the two sides of equal length. 
• All of the sides of a scalene triangle are of different side lengths. It has all angles of different measurements in the same way.

Triangle Characteristics

• All geometrical forms have various side and angle features that help us recognize them. Below is a list of the most important qualities of a triangle.
• Three sides, three vertices, and three internal angles make up a triangle.
• The total of a triangle’s three internal angles is always 180 degrees, according to the angle sum property.
• According to the Triangle Inequality Theorem, the length of the two sides of a triangle is higher than the length of the third side.
• Pythagoras’ theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, i.e. (Hypotenuse2 = Base2 + Altitude2).
• The view from the outside the exterior angle of a triangle is always equal to the sum of the interior opposing angles, according to the triangle theorem.

Formulas in Triangles

Every two-dimensional shape (2D shape) has two basic measures that must be determined in geometry, namely the area and perimeter of that shape. As a result, there are two fundamental formulae for calculating the area and perimeter of a triangle.

• The Triangle’s Perimeter: The perimeter of a triangle is equal to the sum of its three sides. Examine the triangle below, which demonstrates that the perimeter of a triangle is equal to the total of its sides.
• The Triangle’s Area: The area of a triangle is the amount of space it covers. It is equal to half of the sum of its base and altitude (height). Because it is two-dimensional, it is always measured in square units.

Congruence and similarity

If every angle in one triangle has the same measure as the corresponding angle in the other triangle, the two triangles are said to be comparable. The lengths of the respective sides of comparable triangles are in the same proportion, which is also sufficient to prove resemblance.

The following are some basic theorems concerning similar triangles: 

• The triangles are similar if and only if one pair of internal angles of two triangles have the same measure as each other, and another pair has the same measure as each other.
• The triangles are similar if and only if one pair of corresponding sides of two triangles has the same proportion as another pair of corresponding sides, and their included angles have the same measure.
• The triangles are identical if and only if three pairs of matching sides of two triangles are all in the same proportion. Congruent triangles are the same size and shape, all pairs of corresponding interior angles are the same length, and all pairs of corresponding sides are the same length. 

The following are some basic theorems concerning Congruent triangles 

• SAS Postulate: Two sides of a triangle are equal in length to two sides of another triangle, and the included angles are equal in size.
• ASA: In a triangle, two internal angles and the included side have the same measure and length as those in the other triangle.
• SSS: Each triangle’s side is the same length as the matching triangle’s side.
• AAS: In a triangle, two angles and a matching (non-included) side have the same measure and length as in the other triangle.

Point to Remember:  

• A right triangle is defined as a triangle with one right angle.
• An acute triangle is defined as a triangle with three acute angles.
• A right triangle is defined as a triangle with one right angle.
• An obtuse triangle is defined as a triangle with one obtuse angle.
• We call a triangle an equilateral triangle when it has three congruent sides. 
• An equilateral triangle’s angles are always 60 degrees.
• A scalene triangle is a triangle with no congruent sides or angles.