Properties of Triangles | Types, Formulas

I. What is the Shape of Triangle?

Triangles are the simplest polygons in Euclidean geometry, featuring only three sides and three angles. The sides, which are straight lines, connect at three points called vertices. Each vertex forms an angle, and the sum of the three internal angles in a triangle is always 180 degrees. The diagram given below demarcates these terms lucidly.

Lines AB, BC, and CA are the sides of the triangle. Points A, B, and C are the vertices of the triangle. Each vertex contains an internal angle and is denoted as ∠A, ∠B, and ∠C.

As a closed two-dimensional shape, a triangle encloses a region within its boundaries. Hence, a triangle is a convex polygon. Triangles can vary in size and proportions, resulting in different classifications. Despite these differences, all triangles have three sides and three vertices and the most important properties of triangles that we will learn will hold good for all the different types of triangles.

Properties of circles, properties of other polygons such as quadrilaterals, hexagons that we will learn and use while solving quantitative reasoning questions in the GRE will circle back to these important triangle properties.

II. What are the most important Triangle Properties?

1. Sum of Angles (Angle Sum Property):

The sum of the three internal angles of a triangle is always 180 degrees. If the three angles are A, B, and C, then A + B + C = 180

2. Exterior Angle Property of a Triangle

From the angle sum property, we know A + B + C = 180^o
Let’s rewrite the equation as A + B = 180 – C

What is 180 – C? 180 – C is the exterior angle at vertex C.

Before we get around to the property, what is an exterior angle at a vertex?

An exterior angle is formed when one side of a triangle is extended beyond a vertex, creating an angle outside the triangle. In the figure given below, the external angle at vertex C is marked. ∠ACD is the external angle at C.

Note that an angle and its corresponding external angle will add to 180^o. This property applies to any polygon, not just triangles

Now that we know what an exterior angle is, the exterior angle property reads as follows:

We deduced at A + B = 180 – C
Where A and B are the internal angles at the other two vertices and 180 – C is the external angle at vertex C.

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles (the two opposite interior angles).

3. Sum of Exterior Angles of a Triangle

This property applies not only to a triangle, but also to all convex polygons (closed polygons), irrespective of the number of sides that the polygon has.

The sum of the exterior angles of a triangle = 360 degrees.
The sum of the exterior angles of any convex polygion = 360^o

4. Side Lengths Property

Depending upon where you did your schooling the length of sides of a triangle property may have been worded in one of the following ways. Pick the one that you are familiar with. This property is also referred as Triangle Inequality Theorem.

The length of any side of a triangle is always less than the sum of the other two sides and greater than the difference between them.

Reworded version of part 1 of the property: The sum of the lengths of any two sides of a triangle will always be greater than the length of the third side of the triangle.

Let a, b, and c be the measures of the lengths of the three sides of a triangle.

Then, a + b > c and b + c > a and c + a > b

Illustrative Example: We all know 3, 4, and 5 form sides of a triangle (famous Pythagorean triplet). Let’s see whether the side length property holds good in this case.

3 + 4 = 7 > 5

4 + 5 = 9 > 7

3 + 5 = 8 > 4

An easy way to remember this property is by imagining three cities P, Q, and R located on the vertices of a triangle. The direct route from P to Q will always be shorter than traveling from P to R and the from R to Q.

Part 2 of the Side Lengths Property

From the side lengths property we know a + b > c

If we take a to the right side of the inequality, b > c – a

An alternative way to express the Triangle Inequality Property is that the length of any side of a triangle will be greater than the difference between the other two sides.

Or, the difference between any two sides of a triangle will be lesser than the third side of a triangle.

5. Angle – Side Property of a Triangle

The side opposite the largest angle in a triangle is the longest side of the triangle. Extending the logic, the side opposite the smallest angle is the shortest side. 

The three angles of the given Triangle ABC at vertices A, B, and C are 80^o, 60⁰, and 40^o respectively.
80^o is the largest angle in the triangle and the side opposite vertex A, side BC is the longest side.
40^o is the smallest angle in the triangle and the side opposite vertex C, side AB is the shortest side.

The property works even if the anchor is the side.

The angle opposite the longest side is the largest angle in a triangle. Similarly, the angle opposite the shortest side is the smallest angle in the triangle.

i.e., if BC is the longest side of the triangle, the angle in the vertex opposite to BC, which is ∠A will be the largest angle.

III. What are the Types of Triangle?

Triangles can be classified based on two basis. One, types of triangles based on the lengths of the sides of the triangles and the second types of triangles based on the measures of the internal angles of the triangles.

Classification based on the Lengths of the Sides of Triangles

a. Equilateral or Equiangular Triangles

An equilateral triangle is one in which all three sides have equal lengths. In an equilateral triangle all three angles are also equal (60 degrees each). Now it is rather obvious why an equilateral triangle is also called an equiangular triangle.

For instance, if the sides of a triangle measure 10, 10, and 10, the triangle is an equilateral triangle. In general, sides of an equilateral triangle are represented as a, a, and a units.

Because all sides of an equilateral triangle are equal and all angles of an triangle are equal, an equilateral triangle is the regular polygon among triangles. 

b. Isosceles Triangle

Two sides have equal lengths, and the angles opposite these sides are equal. In the above figure, sides AC and BC are equal. The angles opposite side AC, which is ∠B and the one opposite side BC, which is ∠C are equal.

If the sides of a triangle measure, say, 8, 8, and 12 units, the triangle is an isosceles triangle. The sides of an isosceles triangle are generically represented as a, a, and c units.

If the internal angles of a triangle measure, say, 40 degrees, 40 degrees, and 100 degrees, the triangle is an isosceles triangle.

c. Scalene Triangle

All three sides have different lengths, and all three angles are different.

For example, if the sides measure 3, 4, and 5 units, the triangle is a scalene triangle. The triangle has distinct lengths.

Or if the angles of the triangle measured 40 degrees, 60 degrees, and 80 degrees, the triangle is a scalene triangle. The triangle has distinct angles.

Classification based on the Internal Angles of a Triangle

a. Acute Triangle

All three angles of the triangle are acute angles (less than 90 degrees). 

For example, if the internal angles of a triangle measure 40 degrees, 60 degrees, and 80 degrees, the triangle is an acute triangle (also called acute angled triangle).

b. Right Triangle

If one of the internal angles is a right angle (90 degrees), the triangle is a right triangle. 

For example, if the angles of a triangle measure 30 degrees, 60 degrees, and 90 degrees, the triangle is a right triangle. (Because its angles are all different, this one is also a scalene triangle. so, we realize a right angled triangle can be a scalene triangle).

The longest side of a right triangle, the side opposite the largest angle (right angle) is called the hypotenuse. 

In the above figure, angle B is the right angle. A right angle is the largest angle in a right-angled triangle. We have learnt that the side opposite the largest angle is the longest side. The side opposite angle B is AC and is the longest side of the triangle.

Side AC, the longest side of the right triangle is known as the hypotenuse of the triangle. Hypotenuse is always the side opposite the right angle in a right triangle.

All right triangles satisfy the Pythagoras Theorem.

What is the Pythagoras Theorem?

Let the sides of the triangles be a, b, and h, where h is the hypotenuse. ‘a’ and ‘b’ are the lengths of the two perpendicular sides of the triangle.

Pythagoras Theorem states that the sum of the squares of the perpendicular sides equals the square of the hypotenuse.

h² = a² + b²

Special Property Right Angled Triangles

(i) Right Isosceles Triangle

A triangle with one right angle and two equal sides. The angles opposite the two equal sides will measure the same, say x degrees.

By angle sum property, x + x + 90 = 180

So, x = 45

Therefore, the internal angles of a right isosceles triangle are 45 degrees, 45 degrees, 90 degrees.

A right isosceles triangle is also referred to as 45-45-90 triangle.

Ratio of the sides of a Right Isosceles Triangle

The sides opposite the angles 45, 45, and 90 will respectively be in the ratio 1 : 1 : root 2

Let’s quickly check whether the Pythagorean Theorem works in this case

(root 2)² = 1² + 1²

(ii) 30 – 60 – 90 Triangle

With one of the angles being a right angle, the triangle is obviously a right angled triangle.

The sides opposite the angles 30, 60, and 90 degrees will respectively be in the ratio 1 : root 3 : 2

Take a minute to verify whether the sides of the above triangle satisfy the Pythagorean Theorem

c. Obtuse Triangle or Oblique Angled Triangle

If one angle of the triangle is an obtuse angle (greater than 90 degrees), the triangle is an obtuse angled triangle.

For example, if the angles of the triangle measure 30 degrees, 50 degrees, and 100 degrees (obtuse angle), the triangle is an obtuse angled triangle. It is also called oblique angled triangle.

In the above figure, the sides marked in red, AB and BC contain the obtuse angle. And by Side-Angle property, side AC will be the longest side of the triangle.

Possible Triangle Permutations based on the two classifications

	Equilateral	Isosceles	Scalene
Acute Angle Triangle	✓	✓	✓
Right Angled Triangle	✗	✓	✓
Obtuse Triangle	✗	✓	✓

IV. What are the important Triangle Formulas?

A. Area of a Triangle Formulas

The region occupied by a triangle in a two dimensional plane is the area of the triangle. There are at least 5 different formulae to compute the area of a triangle, two of which are useful from the perspective of solving GRE quant questions in geometry.

1. Most important Area of Triangle Formula

Area of a Triangle = 1/2 x Base X Height

Base is the length of any one side of the triangle.

Height is the perpendicular distance between the base and the vertex opposite the base.

2. Heron’s Formula to compute Area of Triangle

a, b, and c are the lengths of the sides of a triangle.

‘s’ is the semi-perimeter of the triangle.

B. Perimeter of Triangles Formula

If the sides of a triangle measure a, b, and c, the perimeter of the triangle is the sum of the three sides of the triangle.

Perimeter of the triangle = a + b + c

C. Semi-Perimeter of Triangles Formula

Semi-perimeter of any triangle is half the perimeter of a triangle.

If the sides of a triangle measure a, b, and c, semi perimeter of the triangle = (a + b + c)/2

Sample Questions in Triangle for exams such as the GRE

Question 1 (GRE Problem Solving Sample Question)

The sides of a triangle measure 5, 10, and x, where x is a positive integer. What is the least possible value of x?

A. 16

B. 14

C. 4

D. 5

E. 6

Approach to Solve the GRE Sample Question in Geometry

Solving the question entails applying the side length property of triangles or the Triangle Inequality Property.

Sum of two sides of the triangle > Third side

For practical purposes of solving questions, the Triangle Inequality Property can be simplified as

Sum of two shorter sides > Longest side.

Why does the above simplification make sense?

If the sum of two smaller numbers is more than the largest number, when we pull the largest number to the left hand side, the inequality will certainly hold good.

We have to find the least possible value of x.

So, it is rather evident that x is not the longest side of the triangle. Between 5 and 10, 5 cannot be the longest side. That leaves us with only option that 10 is the longest side of the triangle.

Sum of 5 and x should be greater than 10.

Apply this logic to find the least possible value of x.

Leave your answer in the comments section of this blog.

Question 2 (GRE Quantitative Comparison Sample Question)

The sides of a triangle measure 10, 10, and root 200 units. 

Quantity A: Numerical value of the area of the triangle.
Quantity B: Numerical value of the semi perimeter of the triangle.

Approach to solve this GRE Quantitative Comparison Question

Identify the type of triangle based on the measure of the lengths of the triangle. This is a crucial step and will help a great deal in computing the area quickly.

If you missed this point, you will have gone about finding the area of the triangle using the Heron’s Formula, which would have been very tedious.

Computing the semi-perimeter is rather straight forward. Approximate the value of root 200 based on finding the square root of the the nearest perfect square to 200. 

Then compare the two quantities. 

Leave your answer in the comments section of this blog.

Summary of the key Properties of Triangles

 Let’s quickly recap all the important Properties of Triangle

1. A convex polygon that has three sides and three angles formed at the three vertices is a triangle.

2. Angle Sum Property: Sum of the internal angles of a triangle equals 180 degrees.

3. External Angle Property: The external angle in any vertex equals the sum of the other two internal angles of a triangle.

4. Side Length or Triangle Inequality Property: Sum of any two sides of a triangle will be greater than the third side.

Or the corollary, the difference between any two sides of a triangle will be lesser than the third side of the triangle.

5. Side Angle Property: The side opposite the largest angle in a triangle is the longest side.

The angle opposite the longest side of a triangle is the largest angle.