Question

Tell whether the given measures could be the angle measures of a triangle. \(90^{\circ}, 90^{\circ}, 90^{\circ}\)

Short answer

No, the sum is 270 degrees, not 180 degrees.

Step-by-step solution

Understand the Properties of Triangle Angles

The sum of the interior angles of any triangle is always 180 degrees. This is a fundamental property of triangles that must be satisfied.

Calculate the Sum of Given Angles

Add the given angle measures: 90 degrees + 90 degrees + 90 degrees.

Verify the Sum

Calculate the sum: \[90^{\text{°}} + 90^{\text{°}} + 90^{\text{°}} = 270^{\text{°}}\] The sum is 270 degrees.

Compare with Triangle Property

Compare the sum of the given angles (270 degrees) with the required sum (180 degrees). Since 270 degrees is not equal to 180 degrees, the given measures cannot be the angles of a triangle.

Key concepts

Properties of Triangles

Triangular shapes are a fundamental part of geometry. One of the most crucial properties of triangles is related to their angle measures.
The sum of all interior angles in any triangle will always be 180 degrees. This is known as the **Triangle Angle Sum Property**. Understanding this helps in identifying valid triangles.

• Equilateral Triangles: All angles are 60 degrees each.
• Isosceles Triangles: Two angles are equal, adding up with the third to make 180 degrees.
• Scalene Triangles: All angles are different but still total 180 degrees.

These properties are not just true by observation. They are mathematically proven and are essential in solving various problems in geometry. Always remember, if the sum doesn't equal 180 degrees, it cannot be a triangle.

Angle Measures

When evaluating angle measures of a given triangle, it is essential to follow certain steps meticulously.
Firstly, identify the angles provided and add them together. For instance, if you are given 90 degrees, 90 degrees, and 90 degrees:

• Step 1: Sum the angles: \(90^{\text{°}} + 90^{\text{°}} + 90^{\text{°}}\)
• Step 2: Calculate: \[90^{\text{°}} + 90^{\text{°}} + 90^{\text{°}} = 270^{\text{°}}\]

After calculating the sum, compare it with 180 degrees.
If it equals 180 degrees, then the provided angles can indeed form a triangle. Otherwise, as in our example with a sum of 270 degrees, it is clear these angles cannot form a triangle. Always verify the angles to understand whether they satisfy the triangle's property requirements.

Interior Angles

Interior angles are the angles inside a triangle. Each triangle has exactly three interior angles.
These angles always combine to form a sum of 180 degrees. Interior angles help determine the type of triangle.

• An **acute triangle** has all interior angles less than 90 degrees.
• An **obtuse triangle** has one interior angle greater than 90 degrees.
• A **right triangle** has one interior angle exactly equal to 90 degrees.

Besides checking the sum of angles, the nature of individual interior angles verifies the type of triangle. For example, three 90-degree angles sum to 270 degrees, thus invalidating it as a triangle. Knowing these angles and their relationships helps in solving problems and verifying triangle properties accurately.