Triangle Theorem

A triangle is a closed polygon having three sides, vertices and angles. PQR is the symbol for a triangle having three vertices, P, Q, and R. Signboards and sandwiches in the shape of a triangle are two of the most common instances of triangles.

the definition of a triangle?

A triangle is a three-sided closed polygon with three interior angles. It is one of the most fundamental shapes in geometry, consisting of three vertices linked together and symbolized by the symbol △. The sides and angles of a triangle are used to classify it into several sorts.

Theorems of Triangles

There are various sorts of triangles based on the length of the sides, such as the scalene triangle, isosceles triangle, and equilateral triangle, as well as triangles depending on the degree of the angles, such as the acute angle triangle, right-angled triangle, and obtuse angle triangle.

Despite the fact that there are several Geometry Theorems on Triangles, let us look at some basic geometry theorems:

• Theorem 1: The total of the three interior angles in any triangle is 180 degrees.

• Theorem 2: When a triangle side is constructed, the exterior angle formed is equal to the sum of the interior opposite angles.

• Theorem 3: The base angles of an isosceles triangle are equivalent.

• Theorem 4: A line is parallel to the third side of a triangle if it divides any two sides of a triangle in the same ratio.

• Theorem 5: If comparable angles are equal in two triangles, then their corresponding sides have the same ratio, and the two triangles are identical.

• Theorem 6: If the sides of one triangle are proportional to the second triangle’s sides, then the corresponding angles are equal, and the two triangles are identical.

Inequality in the Triangle

The triangle inequality theorem is a fundamental mathematical idea that can be found in many different disciplines of mathematics. The educated Civil engineers use the triangle inequality theorem in the real world because their work involves surveying, transportation, and urban planning. They can use the triangle inequality theorem to calculate unknown lengths and get a rough approximation of various dimensions using the triangle inequality theorem.

Triangle Inequality and its significances

According to the Triangle Inequality (theorem), the total of any two sides in a triangle must be greater than the third side. Take, for example, the following triangle ABC:

The Triangle Inequality Theorem states:

AB + BC has to be greater than AC, or AB + BC > AC.

AB + AC must be greater than BC, or AB + AC > BC.

BC + AC has to be greater than AB, or BC + AC > AB.

• Proof of the theorem:

Extend BA to point D, such that AD = AC, and connect C and D.

In BDC Triangle

We can see that ∠ACD = ∠D, which suggests that  in ∆ BCD, ∠BCD > ∠D. As a result, the sides on the opposite side of bigger angles are larger, and thus: BC < BD

BC < AB + AD

BC <  AB + AC (due to the fact that AD = AC)

Thus proved

In the form of a triangle, we can also conclude:

• The difference between any two sides will be less than the third since the sum of any two sides is bigger than the third.

• The sum of any two sides is always greater than the sum of the 3rd  side.

• The longest side in a triangle is the side opposite a greater angle.

The theorem of Triangle Congruence

The triangle congruence theorem is made up of five theorems that show that two triangles are congruent. Two triangles are considered to be congruent or the same if their shape and size are the same, i.e. the matching sides and angles are put in the same location in both triangles.

What is the Triangle Congruence Theorem?

The triangle congruence theorem, often known as the triangle congruence criteria, aids in determining whether or not a triangle is congruent. The phrase congruent refers to objects that are identical in shape and size regardless of how they are turned, flipped, or rotated. Congruent figures are defined as shapes that are overlaid on each other in geometry; for example, triangles and quadrilaterals can be congruent.

The triangle congruence theorems or triangle congruence criteria that facilitate to prove triangle congruence are listed below.

• SSS (Side, Side, Side)

• SAS (Side, Angle, Side)

• ASA (Angle, Side, Angle)

• AAS (Angle, Angle, Side)

• RHS (Right Angle-Hypotenuse-Side)

Conclusion:

A triangle is a three-edged, three-vertices polygon. In geometry, it is one of the fundamental shapes. Triangle ABC refers to a triangle with the vertices A, B, and C. When three points are non-collinear in Euclidean geometry, they form a unique triangle and plane (i.e. a two-dimensional Euclidean space). To put it another way, the triangle exists only in one plane, and every triangle exists in some plane.