Learn About Triangles and Their Properties with This Free PDF Guide

All Types of Triangles and Their Properties PDF Download

Triangles are one of the most basic and common shapes in geometry. They have three sides, three angles, and three vertices. But did you know that there are different types of triangles based on their sides and angles? And did you know that each type of triangle has its own properties and characteristics? In this article, we will explore all types of triangles and their properties in detail. We will also provide you with a PDF file that you can download for free and use as a handy reference.

all types of triangles and their properties pdf download

Triangles are classified into two main groups based on their sides and angles. These are:

• Types of triangles based on sides

• Types of triangles based on angles

Each group has three subtypes, making a total of six types of triangles. We will look at each type of triangle one by one and learn about their definitions, properties, and examples.

Triangles are important because they have many applications in mathematics, science, engineering, architecture, art, and more. They are used to measure distances, angles, heights, areas, volumes, and other quantities. They are also used to construct polygons, polyhedra, circles, spheres, and other shapes. They are also used to model natural phenomena such as light rays, sound waves, forces, and motions.

Types of Triangles Based on Sides

The first group of triangles is based on the lengths of their sides. There are three types of triangles in this group:

• Equilateral triangle

• Isosceles triangle

• Scalene triangle

Equilateral Triangle

An equilateral triangle is a triangle that has all three sides equal in length. It is also called an equiangular triangle because all three angles are equal in measure.

Some properties of an equilateral triangle are:

• All three sides are congruent (equal in length).

• All three angles are congruent (equal in measure).

• All three angles measure 60 degrees.

• The altitude, median, angle bisector, and perpendicular bisector of any side are the same line and divide the triangle into two congruent right triangles.

• The area of an equilateral triangle is given by the formula $$A = \frac\sqrt34s^2$$ where $$s$$ is the length of a side.

• The perimeter of an equilateral triangle is given by the formula $$P = 3s$$ where $$s$$ is the length of a side.

An example of an equilateral triangle is shown below:

/\ / \ / \ /______\

Isosceles Triangle

An isosceles triangle is a triangle that has two sides equal in length. The third side is called the base and the two equal sides are called the legs. The angle opposite to the base is called the vertex angle and the angles opposite to the legs are called the base angles.

Some properties of an isosceles triangle are:

• Two sides are congruent (equal in length).

• Two angles are congruent (equal in measure).

• The base angles are congruent (equal in measure).

• The altitude, median, and angle bisector of the base are the same line and divide the triangle into two congruent right triangles.

• The area of an isosceles triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of the base and $$h$$ is the height.

• The perimeter of an isosceles triangle is given by the formula $$P = 2l + b$$ where $$l$$ is the length of a leg and $$b$$ is the length of the base.

An example of an isosceles triangle is shown below:

/\ / \ / \ /______\ l l

Scalene Triangle

A scalene triangle is a triangle that has no sides equal in length. It is also called an unequal triangle because all three sides are different in length.

Some properties of a scalene triangle are:

• No sides are congruent (equal in length).

• No angles are congruent (equal in measure).

• The area of a scalene triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of any side and $$h$$ is the height corresponding to that side.

• The perimeter of a scalene triangle is given by the formula $$P = a + b + c$$ where $$a$$, $$b$$, and $$c$$ are the lengths of the three sides.

An example of a scalene triangle is shown below:

/\ / \ / \ /______\ a c b

Types of Triangles Based on Angles

The second group of triangles is based on the measures of their angles. There are three types of triangles in this group:

• Acute triangle

• Right triangle

• Obtuse triangle

Acute Triangle

An acute triangle is a triangle that has all three angles acute, that is, they measure less than 90 degrees. It is also called an acute-angled triangle.

Some properties of an acute triangle are:

• All three angles are acute (less than 90 degrees).

• The sum of any two angles is greater than the third angle.

• The longest side is opposite to the largest angle.

• The shortest side is opposite to the smallest angle.

• The area of an acute triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of any side and $$h$$ is the height corresponding to that side.

• The perimeter of an acute triangle is given by the formula $$P = a + b + c$$ where $$a$$, $$b$$, and $$c$$ are the lengths of the three sides.

An example of an acute triangle is shown below:

/\ / \ / \

a c b

Right Triangle

A right triangle is a triangle that has one angle that measures exactly 90 degrees. It is also called a right-angled triangle. The side opposite to the right angle is called the hypotenuse and the other two sides are called the legs.

Some properties of a right triangle are:

• One angle is right (90 degrees).

• The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This is known as the Pythagorean theorem and can be written as $$a^2 + b^2 = c^2$$ where $$a$$ and $$b$$ are the lengths of the legs and $$c$$ is the length of the hypotenuse.

• The area of a right triangle is given by the formula $$A = \frac12ab$$ where $$a$$ and $$b$$ are the lengths of the legs.

• The perimeter of a right triangle is given by the formula $$P = a + b + c$$ where $$a$$, $$b$$, and $$c$$ are the lengths of the three sides.

An example of a right triangle is shown below:

/ / / /___ a c b

Obtuse Triangle

An obtuse triangle is a triangle that has one angle that measures greater than 90 degrees. It is also called an obtuse-angled triangle or an oblique triangle.

Some properties of an obtuse triangle are:

• One angle is obtuse (greater than 90 degrees).

• The sum of any two angles is less than the third angle.

• The longest side is opposite to the obtuse angle.

• The shortest side is opposite to the smallest angle.

• The area of an obtuse triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of any side and $$h$$ is the height corresponding to that side.

• The perimeter of an obtuse triangle is given by the formula $$P = a + b + c$$ where $$a$$, $$b$$, and $$c$$ are the lengths of the three sides.

An example of an obtuse triangle is shown below:

/\ / \ / \ /______\ c b a

Types of Triangles Based on Sides and Angles

The third group of triangles is based on both their sides and angles. There are six types of triangles in this group:

• Equiangular triangle

• Isosceles right triangle

• Obtuse isosceles triangle

• Acute isosceles triangle

• Right scalene triangle

• Obtuse scalene triangle

Equiangular Triangle

An equiangular triangle is a triangle that has all three angles equal in measure. It is also called an equilateral triangle because all three sides are equal in length.

Some properties of an equiangular triangle are:

• All three angles are congruent (equal in measure).

• All three angles measure 60 degrees.

• All three sides are congruent (equal in length).

• The altitude, median, angle bisector, and perpendicular bisector of any side are the same line and divide the triangle into two congruent right triangles.

• The area of an equiangular triangle is given by the formula $$A = \frac\sqrt34s^2$$ where $$s$$ is the length of a side.

• The perimeter of an equiangular triangle is given by the formula $$P = 3s$$ where $$s$$ is the length of a side.

An example of an equiangular triangle is shown below:

/\ / \ / \ /______\

Isosceles Right Triangle

An isosceles right triangle is a triangle that has two sides equal in length and one angle that measures 90 degrees. The two equal sides are called the legs and the third side is called the hypotenuse.

Some properties of an isosceles right triangle are:

• Two sides are congruent (equal in length).

• Two angles are congruent (equal in measure).

• One angle is right (90 degrees).

• The base angles are congruent (equal in measure) and measure 45 degrees.

• The hypotenuse is equal to the product of the leg and $$\sqrt2$$.

• The area of an isosceles right triangle is given by the formula $$A = \frac12l^2$$ where $$l$$ is the length of a leg.

• The perimeter of an isosceles right triangle is given by the formula $$P = 2l + l\sqrt2$$ where $$l$$ is the length of a leg.

An example of an isosceles right triangle is shown below:

/ / / /___ l l

Obtuse Isosceles Triangle

An obtuse isosceles triangle is a triangle that has two sides equal in length and one angle that measures greater than 90 degrees. The two equal sides are called the legs and the third side is called the base.

Some properties of an obtuse isosceles triangle are:

• Two sides are congruent (equal in length).

• Two angles are congruent (equal in measure).

• One angle is obtuse (greater than 90 degrees).

• The base angles are congruent (equal in measure) and measure less than 45 degrees.

• The area of an obtuse isosceles triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of the base and $$h$$ is the height.

• The perimeter of an obtuse isosceles triangle is given by the formula $$P = 2l + b$$ where $$l$$ is the length of a leg and $$b$$ is the length of the base.

An example of an obtuse isosceles triangle is shown below:

/\ / \ / \ /______\ l l

Acute Isosceles Triangle

An acute isosceles triangle is a triangle that has two sides equal in length and all three angles acute, that is, they measure less than 90 degrees. The two equal sides are called the legs and the third side is called the base.

Some properties of an acute isosceles triangle are:

• Two sides are congruent (equal in length).

• Two angles are congruent (equal in measure).

• All three angles are acute (less than 90 degrees).

• The base angles are congruent (equal in measure) and measure more than 45 degrees.

• The area of an acute isosceles triangle is given by the formula $$A = \frac12bh$$ where $$b$$ is the length of the base and $$h$$ is the height.

• The perimeter of an acute isosceles triangle is given by the formula $$P = 2l + b$$ where $$l$$ is the length of a leg and $$b$$ is the length of the base.

An example of an acute isosceles triangle is shown below:

/\ / \ / \ /______\ l l

Right Scalene Triangle

A right scalene triangle is a triangle that has one angle that measures 90 degrees and no sides equal in length. The side opposite to the right angle is called the hypotenuse and the other two sides are called the legs.

Some properties of a right scalene triangle are:

• No sides are congruent (equal in length).

• No angles are congruent (equal in measure).

the unknown angle.

• For a right triangle, you can use the fact that one angle is 90 degrees and the other two angles are complementary. You can write an equation like $$A + B = 90$$ where $$A$$ and $$B$$ are the acute angles of the triangle and solve for the unknown angle.

• For a triangle with known side lengths, you can use the law of cosines to find any angle. The law of cosines states that $$c^2 = a^2 + b^2 - 2ab\cos C$$ where $$a$$, $$b$$, and $$c$$ are the lengths of the three sides and $$C$$ is the angle opposite to $$c$$. You can rearrange this equation to find $$\cos C$$ and then use a calculator to find $$C$$.

Q5: How do you draw a triangle?

A5: There are different ways to draw a triangle depending on the given information. Some common ways are:

• For a triangle with known side lengths, you can use a ruler and a compass to draw the triangle. You can start by drawing one side with the ruler and then use the compass to mark the lengths of the other two sides from the endpoints of the first side. Then you can connect the marks to form the triangle.

• For a triangle with known angles, you can use a protractor and a ruler to draw the triangle. You can start by drawing one side with the ruler and then use the protractor to measure the angles from the endpoints of the first side. Then you can draw the other two sides with the ruler to form the triangle.

• For a right triangle with known leg lengths, you can use a ruler and a set square to draw the triangle. You can start by drawing one leg with the ruler and then place the set square along that leg. Then you can draw the other leg with the ruler along the edge of the set square. Then you can connect the endpoints of the legs to form the triangle.

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