Properties of Triangles

A closed curve formed by three line segments is called a triangle. There are primarily three sides, three angles, and three verticals of a triangle. 

For example, in an ABC triangle, signified by ∆ABC, AB, BC, CA are the sides, ∠A, ∠B, ∠C are the angles, and A, B, C are the verticals. 

The two basic ways to classify a triangle are considering its sides and angles.  

Types of Triangle

It is essential to know about the different types of triangles to understand the properties of triangles. 

Based on Sides

Equilateral Triangle

The equilateral triangle is a triangle that has all three sides of equal lengths, and all three angles have equal estimates of 60°. 

Example: ∠A = ∠B = ∠C

Isosceles Triangle

The isosceles triangle is a triangle that has two sides of equal length, and the two angles opposite the equal sides are equal themselves. 

Example: ∠B = ∠C

Scalene Triangle

The scalene triangle is a triangle that has all three sides of different lengths, and all three angles have different estimates. 

Example: ∠A ≠ ∠B ≠ ∠C

Based on Angles

Acute Angle Triangle

The acute-angle triangle is a triangle in which all three angles are less than 90°. Here, the sum of the square of any two sides is more than the square of the third side.

Example: ∠A < 90°, ∠B < 90°, ∠C < 90°

Right Angle Triangle

The right-angle triangle is a triangle in which one of the angles is 90°. Pythagoras theorem is applicable on this triangle, where the sum of the square of two sides is equal to the square of the third side, which is opposite to 90°. 

Example: ∠B = 90°

Obtuse Angle Triangle

The obtuse-angle triangle is a triangle in which one of the angles is more than 90°. In this triangle, the sum of the square of two smaller sides is less than the square of the largest side. 

Example: ∠A > 90°

Properties of Triangles

The basic properties of the triangle are given below.

Angle Sum Property

The angle sum property expresses that the sum of the measures of the three angles of a triangle is 180°.

Exterior Angle Property

The exterior angle property expresses that an exterior angle of a triangle is always equal to the sum of opposite interior angles. 

Sum of Lengths of Sides Property

The sum of lengths of any two sides of a triangle is always more than the length of the third side. Therefore, three line segments whose lengths do not satisfy this property cannot form a triangle. 

Segments of Triangles 

Let us discuss the three segments in a triangle.

Median of Triangle

The median is a line segment on the triangle. It has one endpoint at the vertex and another at the midpoint of the opposite side of the triangle. There are three medians of every triangle.

Altitude of Triangle

It is the line segment extracted from a vertex perpendicular to the opposite side. It is basically the extension of that side of the triangle. Most altitudes derive from the base, and their length is also the triangle’s height. There are three altitudes of every triangle. 

Angle Bisector

The line segment that bisects an angle of the triangle is the bisector. It intersects the opposite side of the triangle. There are three bisectors of every triangle. The theorem of angle bisectors applies to all types of triangles.  

Perpendicular Bisector

As the name suggests, this line segment is perpendicular to the triangle. It passes through the midpoint of a side of the triangle. Perpendicular bisectors do not always have to bisect the opposite vertex.  

Congruent Triangles

The rules of congruency of the triangles are as below.

SSS Congruency

The rule of SSS congruence asserts that two triangles are congruent if all three sides of one triangle are equivalent to the corresponding sides of the second triangle.

SAS Congruency

As per this rule, two triangles are congruent if the two sides and the included angle of one triangle are equivalent to the two sides and the included angle of the second triangle. 

ASA Congruency

The rule of ASA congruence states that two triangles are congruent if the two angles and the included side of one triangle are equivalent to the two angles and the included side of the second triangle. 

AAS Congruency 

According to AAS congruency, two angles are congruent if the two angles and the non-included side of one triangle are equivalent to the two angles and the non-included side of the second triangle. 

RHS Congruency

The rule of RHS congruence asserts that two right triangles are congruent if the hypotenuse and one side of one triangle are equivalent to the hypotenuse and corresponding side of the second triangle.

Conclusion

A polygon having three sides and three angles is a triangle. The triangle is considered a significant part of mathematics. Being a polygon, the three sides and angles of a triangle demonstrate some basic properties. 

In this topic, we discussed the types of triangles, properties of triangles, and other essential miscellaneous topics in detail. You can also check out some relevant concepts and rules of the triangle to get a holistic understanding of the topic.