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Shapes and patterns have always fascinated us since the dawn of time. The rigidness in some and swirls in others have often puzzled our minds and nature housing such beautiful designs is beyond fascinating. Triangles are one of those shapes that possess a variety of features and applications in our world. A triangle is defined as a closed two-dimensional figure with 3 sides, 3 angles, and 3 vertices in Geometry. They are a type of polygon whose sum of all three angles usually equals 180°.

Since a triangle is a two-dimensional shape, the types of triangles are classified based on their sides and angle measurement.

A triangle is classified into three types based on the length of its sides,

Scalene Triangle – The length of all sides differs or is not equal.

Isosceles Triangle – The properties of an isosceles triangle involve a triangle having two sides equal in length and the third is not equal.

Equilateral Triangle – The properties of an equilateral triangle involve a triangle having the length of all three sides equal.

A triangle is classified based on the angles,

Acute Angle Triangle – The angles of a triangle are less than 90°

Obtuse Angle Triangle – One angle of a triangle is greater than 90°

Right Angle Triangle – One angle of a triangle is equal to 90°

Triangles follow certain properties and rules to achieve a particular state and to solve complex geometric problems. Some of the properties of triangles are listed below,

The presence of three sides, three angles, and three vertices.

All the interior angles equal to 180°.

The third side of a triangle is less than the sum of the other two sides.

The half product of the base and the height gives the area of the triangle.

The sum of all the three sides of a triangle provides the perimeter of the triangle.

The following properties are established to find the median of a triangle,

A triangle has 3 medians, one from each vertex.

All medians meet at a single point.

The Centroid of the triangle is the point where the 3 medians meet.

The median of a triangle divides it into two smaller triangles.

When all three sides and three angles of a given triangle are equal it is referred to as a congruent triangle. It is based on the shape and size of the triangle. The object and its mirror image are generally referred to as congruence. Two images are congruent if they superimpose each other. In geometric figures, a similar length of line segments are congruent and so is it’s the angle of measurement.

Conditions for a Congruent Triangle are established as the property of sides of a triangle.

### SSS (Side-Side-Side)

All three sides are equivalent concerning the second triangle.

### SAS (Side-Angle-Side)

Any two sides and an angle are equivalent concerning the second triangle.

### ASA (Angle-Side-Angle)

Any two angles and a side are equivalent concerning property f a second triangle.

### AAS (Angle-Angle-Side)

A non-included side and two angles are equal to corresponding angles and sides of another triangle.

### RHS (Right Angle-Hypotenuse-Side)

The hypotenuse with a side of a right-angled triangle is equivalent to the second triangle’s hypotenuse and right-angled side.

In the figure, ΔABC and ΔPQR are congruent triangles. Therefore,

Vertices: A and P, B and Q, C and R vertices are equal

Sides: AB=PQ, QR= BC, and AC=PR;

Angles: ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R

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The Angle Sum Property of a given triangle is defined as the sum of all the interior angles of a triangle is equal to 180°.

Devise a theorem to calculate the angle sum property of a triangle.

Let’s take a ΔABC , prove the property of the triangle by drawing a line PQ,

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Since PQ is a straight line,

∠PAB + ∠BAC + ∠QAC = 180⁰

PQ ll BC and AB, AC are transversals

Hence, ∠QAC = ∠ACB and ∠PAB = ∠CBA are a pair of alternate angles

Substituting ∠QAC and ∠PAB , we get

∠ACB + ∠BAC + ∠CBA = 180⁰

Hence, the sum of all interior angles is equal to 180⁰.

FAQ (Frequently Asked Questions)

1.How to Find the Area of a Triangle and What is Heron’s Formula?

The area of a triangle establishes the region enclosed by a three-sided polygon in one plane. The equation for the area of a triangle is given as,

Area of a Triangle = ½ x base x height

An additional modification was made to the above formula. The new formula includes the lengths of the sides of a given triangle. This formula is termed as Heron’s formula and is stated below,

A = √s(s - a)(s - b)(s - c)

Here, ‘s’ is the semi-perimeter of the triangle and is described as,

s = (a + b + c)/2

Here, a, b and c are the measures of the sides of a given triangle.

2.How to Find the Perimeter of a Triangle and What are Its Applications?

The perimeter of a triangle is formulated by including the sum of the lengths of all sides of the triangles existing in one plane. Usually, in the field of architecture, the perimeter is used to construct blueprints of building plans and structures. The formula for the perimeter of a given triangle is as follows,

P = a + b + c

In the field of Mathematics, the triangle is used to study various levels of geometry and in a variety of planes. The perimeter calculated for a triangle in one plane differs from its existence in a second plane on a different axis. Therefore, it is important to consider all possible planes of existence and to keep in mind that a triangle is a two-dimensional shape.