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Chapter 13: Problem 41

Classify a triangle with an angle of \(125^{\circ}\) as acute, obtuse, or right.

Short Answer

Expert verified
The triangle is obtuse.

Step by step solution

01

Understand Triangle Types by Angles

There are three main types of triangles classified by their angles: acute triangles (all angles are less than 90掳), right triangles (one angle is exactly 90掳), and obtuse triangles (one angle is greater than 90掳).
02

Analyze Given Angle

The triangle has been given to have an angle of 125掳. An angle of 125掳 is greater than 90掳.
03

Determine Triangle Type

Since the triangle has an angle greater than 90掳, it fits the definition of an obtuse triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Types
Triangles are fundamental geometric shapes and can be classified by their angles. This classification helps us understand and describe the specific properties of each type of triangle. Here's a quick breakdown:
  • **Acute Triangles**: Every angle in an acute triangle is less than 90掳. This means the triangle is very sharp-angled overall, creating a narrow shape.
  • **Right Triangles**: One of the angles in a right triangle is exactly 90掳, forming a perfect "L" shape on one side. It's one of the simplest and most recognized forms because it follows Euclidean geometry principles.
  • **Obtuse Triangles**: These triangles have one angle greater than 90掳. This makes them appear wider and more "stretched out" when compared to the other types. Each obtuse triangle will only have one obtuse angle, as the total sum of angles in any triangle must be 180掳.
Understanding these classifications is crucial when solving geometric problems, as knowing the type of triangle provides insights into its side lengths, area calculations, and more.
Obtuse Triangles
Obtuse triangles are unique in their appearance and characteristics. Recognizing these shapes involves identifying the presence of an obtuse angle, which is greater than 90掳 but less than 180掳. Features of Obtuse Triangles:
  • They always contain one and only one obtuse angle, making them different from acute and right triangles.
  • The other two angles in an obtuse triangle are always acute, meaning they are less than 90掳, because the sum of all angles in a triangle is 180掳.
  • Visually, obtuse triangles often look more stretched out and have a "flat" appearance on one side.
Such triangles are common in various real-world applications, such as in certain types of architectural designs and road angles.
Angle Measurement
Angles are the building blocks of geometry, and their measurement is essential in classifying triangles. Understanding Angle Measurement:
  • **Degrees**: The most familiar way to measure angles is in degrees. A full circle is 360掳, and hence, angles are measured according to this system.
  • When classifying triangles, each angle is measured, and the values are added together, which must equal exactly 180掳 for a valid geometric triangle.
  • In problems like the given exercise, recognizing a 125掳 angle immediately helps identify the triangle type, as it is greater than 90掳.
Knowing how to measure and interpret angles is a foundational skill in geometry, assisting in more complex problem-solving and analysis.

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