Problem 5 - Algebra and Trigonometry

Question

If none of the angles of a triangle is a right angle, the triangle is called _________. (a) oblique (b) obtuse (c) acute (d) scalene

Step-by-Step Solution

Answer

The correct answer is (a) oblique.

Step 1: Understand the Definition of the Terms

To solve this problem, we need to understand the definitions of different types of triangles. Specifically, we need to know what 'oblique,' 'obtuse,' 'acute,' and 'scalene' mean.

Step 2: Define 'Oblique Triangle'

A triangle that does not have a right angle is called an oblique triangle. It may have either all acute angles or one obtuse angle.

Step 3: Define 'Obtuse Triangle'

An obtuse triangle is one that has one angle greater than 90 degrees. This type of triangle cannot fit the definition provided because it specifies only one obtuse angle.

Step 4: Define 'Acute Triangle'

An acute triangle is a triangle where all three angles are less than 90 degrees. While it matches part of the condition, it is not the most general answer.

Step 5: Define 'Scalene Triangle'

A scalene triangle is one where all three sides are of different lengths. This characteristic has no bearing on the angles themselves and does not answer the question.

Step 6: Choose the Correct Answer

From the provided definitions, it is clear that the term 'oblique' is the most general and correct term for a triangle that does not have a right angle.

Key Concepts

Oblique Triangle

An oblique triangle is any triangle that does not contain a right angle. This means that none of its angles are exactly 90 degrees. Oblique triangles can be further classified into two categories:

• Acute Triangle: All three angles are less than 90 degrees.
• Obtuse Triangle: One of the angles is greater than 90 degrees.

Understanding oblique triangles helps in distinguishing them from right triangles, which have one angle exactly equal to 90 degrees. So, whenever you encounter a triangle without a right angle, you can confidently call it an oblique triangle.

Obtuse Triangle

An obtuse triangle features one angle that is greater than 90 degrees but less than 180 degrees. This makes it stand out from acute and right triangles. Here are some key points:

• Single Obtuse Angle: Only one angle is obtuse, while the other two are acute (less than 90 degrees).
• Sum of Angles: The sum of all angles is still 180 degrees, just like any other triangle.

Remember, an obtuse triangle cannot have more than one obtuse angle, as that would exceed the total angle sum of 180 degrees.

Acute Triangle

In an acute triangle, all three internal angles are less than 90 degrees. This gives the triangle a distinct look where all angles are sharp and pointed. Key attributes include:

• All Acute Angles: Each of the three angles is less than 90 degrees.
• Angle Sum: The sum of angles still equals 180 degrees.

Acute triangles are easy to identify because every angle within them is visibly smaller than a right angle.

Scalene Triangle

A scalene triangle is unique because all three sides are of different lengths, and consequently, all three angles are different as well. Important features to note are:

• Different Side Lengths: No two sides are the same length.
• Different Angles: Each angle has a different measure.

While the concept of a scalene triangle focuses primarily on the lengths of sides, this variation in side lengths also leads to different angles, making it a versatile concept in geometry.