**TRIANGLES AND ANGLES OF A TRIANGLE**

**TRIANGLES:**

Triangles are two-dimensional shapes made up of three sides and three vertices. In spite of the fact that not every one of them is the same, they all must have two components set up so as to be a triangle.

**A triangle is a closed figure made of three line segments.**

A triangle needs to have three line segments and three angles. It is the geometric shape formed by the lowest number of sides and angles.

In the above triangle, AB, BC, CA are the three line segments and ∠A, ∠B, ∠C are the three angles of ∆ABC.

We can call a triangle as a polygon, with three sides, three angles, and three vertices.

**TYPES OF TRIANGLES:**

Triangles can be classified in 2 ways, according to angles and according to length of the sides.

According to angles, there are three types of triangles – acute, right, and obtuse-angled triangle.

According to length of sides, triangles can be classified as – Scalene, Isosceles, and Equilateral triangle.

**The types of Triangle based on Sides are:**

- EQUILATERAL TRIANGLE:

A triangle having all the three sides of equivalent length is called an equivalent triangle.

If we divide 180º by the triangle’s number of sides to determine the three angles’ measurements, we will get 60º. Each of the equilateral triangle’s angle measures 60º.

- ISOSCELES TRIANGLE:

A triangle having any 2 sides of equivalent length is an Isosceles triangle.

The two opposite angles of equal sides are also equal in an isosceles triangle. The side that is different is located precisely between the equal angles.

- SCALENE TRIANGLE:

A triangle having three sides of different lengths is known as a scalene triangle.

**The types of Triangle based on Angles are:**

- ACUTE ANGLED TRIANGLE:

A triangle whose all angles are less than 90 degrees is called an acute angled triangle or Acute triangle.

**Note:** The equilateral triangle is also acute since all of its angles measure 60º. Isosceles and scalene triangles can also be acute if all of their angles are acute.

- OBTUSE ANGLED TRIANGLE

Obtuse angled triangle: A triangle whose one angle is greater than 90 degrees is called obtuse angled triangle or obtuse triangle.

**Note:** In obtuse triangles, one angle is obtuse. The other two are acute. If the acute angles are equal, the obtuse triangle will also be isosceles. If all of the angles are different, the triangle will be scalene.

- RIGHT-ANGLED TRIANGLE

Right-angled triangle: A triangle whose any one angle is of 90 degrees is a Right-angled triangle or Right triangle.

In the figure over, the side opposite is right angle, BC is known as the hypotenuse.

For a Right triangle ABC,

BC^{2} = AB^{2} + AC^{2}

This is known as the Pythagorean Theorem.

In the above triangle we have, 5^{2} = 4^{2} + 3^{2}.

Any triangle that fulfills this condition is a right triangle.

Consequently, the Pythagorean Theorem assists with discovering whether a triangle is Right-calculated.

**Note:**

- The right angle measures 90º and the triangle has three angles that add up to 180º, hence the other two angles are acute and add up to 90º.
- A right triangle can be isosceles if its two acute angles both measure 45º.
- If all angles have different measurements, the triangle will be scalene.

Browse the video and learn by visually the concept of triangles.

**Types of Right angled triangle:**

There are various sorts of right triangles. Starting at now, our emphasis is just on a unique pair of right triangles.

**ISOSCELES RIGHT TRIANGLE**: 45-45-90 triangle

A 45-45-90 triangle is a right triangle in which the other two angles are 45° each.

This is an example of a isosceles right triangle.

In ∆ DEF, we have DE = DF and ∠D = 90°.

The sides in this triangle are in the ratio 1 : 1 : √2.

**SCALENE RIGHT TRIANGLE**: 30-60-90 triangle:

A 30-60-90 triangle, as the name demonstrates, is a right triangle whose other two angles are 30° and 60°.

This is a scalene right triangle as none of the sides or points are equivalent.

The sides in a 30-60-90 triangle are in the proportion 1 : √3 : 2

Like these two triangles satisfy the Pythagorean Theorem.

**PROPERTIES OF TRIANGLES:**

- The sum of the three angles in a triangle is 180°.

**This is known as the Angle Sum Property of Triangle.**

- The sum of the lengths of any two sides of a triangle is more than the length of the third side.
- The side opposite to the longest angle is the longest side of the triangle and the side opposite to the smallest angle is the shortest side of the triangle.

In the figure above, ∠B is the biggest edge and the side inverse to it (hypotenuse), is the biggest side of the triangle.

In case of a right-angled triangle, the largest side is called its hypotenuse

In the figure above, ∠A is the largest angle and the side opposite to it, BC is the longest side of the triangle.

- An exterior angle of a triangle is equivalent to the total of its interior opposite angles. This is known as the exterior angle property of a triangle.

Here, ∠ACD is the exterior angle of the ∆ABC.

So, ∠ACD = ∠CAB + ∠ABC. (Due to Exterior Angle Property)

- The sum of all exterior angles of any triangle is equal to 3600.
- The difference between the lengths of any two sides of a triangle is always less than the length of the third side.
- The height of a triangle is equal to the length of the perpendicular dropped from a vertex to its opposite side, and this side is called its base

Now let’s discuss how to solve questions related to Triangles:

**Question: 1**

In an isosceles triangle DEF, if an interior angle ∠D = 1000 then find the value of ∠F?

Solution

Step 1: Giveno

∆DEF is an isosceles triangle

∠D = 1000

Step 2: To find

The value of ∠F

Step 3: Approach

We know that the sum of all interior angles in a triangle = 1800

Hence, ∠D + ∠E + ∠F = 1800

∠E + ∠F = 1800 – 1000 = 800

Since ∆DEF is an isosceles triangle; two of its angles must be equal.

The only possibility is ∠E = ∠F

2∠F = 800

∠F = 400

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