Question

Is it possible to have two obtuse angles in a triangle? Write a few sentences explaining why or why not.

Short answer

No, a triangle cannot have two obtuse angles because their sum would exceed 180 degrees.

Step-by-step solution

Understanding Triangle Angles

A triangle has three angles and they must sum to 180 degrees. This fundamental property is crucial when examining the types of angles a triangle can have.

Definition of an Obtuse Angle

An obtuse angle is one that is greater than 90 degrees. By defining this, we acknowledge that if a triangle has one obtuse angle, it must be greater than 90 degrees.

Calculating Remaining Angle Sum

If a triangle has one obtuse angle, say 110 degrees, the sum of the other two angles must be 70 degrees because 180 - 110 = 70.

Checking Possibility of Two Obtuse Angles

Since an obtuse angle is greater than 90 degrees, having two obtuse angles would mean two angles exceeding 90 degrees. This would make their sum over 180 degrees, which is impossible in a triangle because the sum of all angles in a triangle must be exactly 180 degrees.

Key concepts

Understanding Obtuse Angles in a Triangle

Triangles can have different types of angles, and understanding these is crucial when analyzing their properties. An obtuse angle is a type of angle that measures greater than 90 degrees. In any triangle, the angles play a central role in defining its shape and type.
When considering whether a triangle can have two obtuse angles, we need to remember the properties of triangles. Since an obtuse angle already consumes more than half of the available 180 degrees in a triangle, having two obtuse angles would require more than 180 degrees in total. This is impossible as triangles have a fixed angle sum property, making it impossible for two angles to be obtuse simultaneously.

The Sum of Angles in a Triangle

The sum of the angles in any triangle is always 180 degrees. This is one of the most fundamental properties of triangles and is essential when analyzing the types of angles a triangle can contain.
When you have one obtuse angle in a triangle, which is more than 90 degrees, the remaining two angles must add up to less than 90 degrees. For example, if a triangle has an obtuse angle of 110 degrees, the other two angles together must total 70 degrees because 180 - 110 = 70.

• This ensures that the triangle remains within the universally accepted angle sum property of 180 degrees.
• Since each angle in a triangle contributes to this total, it's impossible for two angles to both be obtuse, as their individual measures would exceed the total possible sum.

Key Triangle Properties Explained

Triangles are geometric shapes with three sides and three angles. Several properties underlie their geometry, making them a rich area of study in mathematics.
One of the key properties is the sum of its interior angles being exactly 180 degrees, which dictates what the possible angle measures can be. This gives rise to basic classification into acute, right, and obtuse triangles.

• An acute triangle has all angles less than 90 degrees.
• A right triangle has one 90-degree angle.
• An obtuse triangle, interestingly, has precisely one angle greater than 90 degrees, while the others compensate within the 180-degree limit.

Knowing these properties helps in understanding why certain combinations of angles, like having two obtuse angles, are impossible within triangles.