Can a triangle have two right angles?

In recent times, geometry is performing a major role in infrastructural development. Geometry has made its mark in the field of construction, architecture, designing, and even in the IT and computing sectors. Geometrical structures and their commendable use can be seen in our ancient monuments and architectures. The application of various geometrical structures and shapes in a specified way describes that the study of geometry carries a long history. 

The term geometry itself was originally derived from the Greek words ‘ge’ which means earth and ‘materia which means measurement.

Geometry is a discipline of mathematics concerned with study of various shapes and structures along with their properties.

What are geometrical shapes?

Geometrical shapes are the area enclosed figures created with a number of boundaries, curves, lines, etc. Triangles, circles, squares, rectangles are some of the common two-dimensional shapes presented on plane surfaces. Whereas, cube, cuboid, prism are three-dimensional structures.

What is a triangle?

Triangles are the area enclosed geometrical figures consisting of three line segments connected with each other. All three sides of a triangle create a distinct angle with the connected segment. But unlike other geometrical shapes like squares and rectangles, its angles do not measure the same in all cases. These triangles are named according to the measure of angle found in them. For example, a triangle having its one angle of 90° is called a right-angled triangle.

What is a right-angled triangle?

A triangle in which one of the angles is equal to 90° is a right-angled triangle and the sum of the other two angles is 90 degrees. The sides having the 90 degrees angle are considered perpendicular and base. And, the side-lying opposite the 90 degrees is known as hypotenuse. The hypotenuse is the longest side of a right-angled triangle.

The relation between the angles and sides of a right-angled triangle is explained by the Pythagoras theorem.

Pythagoras Theorem

The theorem explains that the hypotenuse of a right-angled triangle is equal to the sum of its perpendicular and base.

Hypotenuse² = Perpendicular² + Base²

Properties of a right-angled triangle

• One angle of a right-angled triangle is always 90degrees.
• The side opposite to the angle 90° is called the hypotenuse.
• The hypotenuse is the longest side of the triangle.
• The two sides adjacent to the right angle are called base and perpendicular.
• The sum of the other two interior angles of the right-angle triangle is equal to 90°.
• The area of the right-angle triangle is equal to half of the product of adjacent sides of the right angle.

Area of Right Angle Triangle = ½ (Base × Perpendicular)

Can a triangle have two right angles?

Answer:

A triangle can never have more than one right angle. a triangle consists of three correspondent sides with interior angles whose sum equals to 180 degrees. If a triangle consists of two right angles then, one of its sides will overlap the other making the third angle measure 0 degrees.

Now, let’s prove the statement mathematically.

Let us consider a right-angled triangle ABC.

Now,

∠B = 90° and △ABC

The sum of all three corresponding angles is 180°.

=>∠A+∠B+∠C=180

=>∠A+∠C=180-90

=>∠A+∠C=90

Hence, the sum of both interior angles will be 90 degrees which means

=>∠A≠0

=>∠C≠0

Hence, a triangle can never have two right angles.

Sample Problems

Problem 1. In a right triangle, if perpendicular = 4cm and base = 3cm, then what is the value of hypotenuse?

Solution: 

Perpendicular = 4cm

Base = 3cm

We need to find the hypotenuse.

By Pythagoras theorem, we know that;

Hypotenuse = √(Perpendicular² + Base²)

=>h = √(16 + 9)

= >√25

=> 5cm

Problem 2. In a right-angled triangle, if hypotenuse=13 and base=12, then what is the value of perpendicular?

Solution:

Hypotenuse = 13cm

Base = 12cm

We need to find the perpendicular.

By Pythagoras theorem, we know that;

Hypotenuse = √(Perpendicular2 + Base2)

=>p = √(13)² -( 12)²

= >√169-144

=>√25

=>5cm

Problem 3. In a right-angled triangle, if hypotenuse=5cm and perpendicular=3cm, then what is the value of base.

Solution:

Hypotenuse = 5cm

perpendicular = 3cm

We need to find the base

By Pythagoras theorem, we know that;

Hypotenuse = √(Perpendicular2 + Base2)

=>b = √(5)² -( 3)²

= >√25-9

=>√16

=>4cm