Problem 57 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 57-59, determine whether the statement is true or false. Justify your answer. If a triangle contains an obtuse angle, then it must be oblique.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the properties of obtuse and oblique triangles

A triangle is classified as an obtuse triangle if it contains one angle that measures more than 90 degrees. A triangle is classified as oblique if it does not contain a right angle, which means it can be either an acute triangle (all angles less than 90 degrees) or an obtuse triangle (one angle more than 90 degrees).

Step 2: Connect the properties to the statement

According to the classifications, all obtuse triangles are also oblique triangles because obtuse triangles do not contain a right angle. Therefore, if a triangle is obtuse, it will automatically be an oblique triangle.

Step 3: Conclusion on the statement

Therefore, the given statement 'If a triangle contains an obtuse angle, then it must be oblique' is true. This is because an obtuse triangle, which by definition contains an obtuse angle, falls into the category of oblique triangles as it does not contain a right angle.

Key Concepts

Obtuse Triangle

An obtuse triangle is a triangle with one angle that measures more than 90 degrees. This type of angle is known as an obtuse angle. In any triangle, the sum of all three angles always equals 180 degrees. Thus, if one angle is obtuse (more than 90 degrees), the other two angles together must be less than 90 degrees.

Here are a few key points about obtuse triangles:

• They have exactly one obtuse angle.
• The other two angles are necessarily acute (less than 90 degrees).
• The side opposite the obtuse angle is the longest side of the triangle.

Understanding these characteristics helps in identifying such triangles and knowing their properties allows easier classification of triangles in geometry.

Oblique Triangle

An oblique triangle is one that does not have a right angle. This means that none of its angles measure exactly 90 degrees. Oblique triangles can be further classified into two types:

• Acute Triangle: All three angles are less than 90 degrees.
• Obtuse Triangle: One angle is more than 90 degrees.

When dealing with oblique triangles, it's important to note:

• They always lack a right angle.
• They can have any combination of angles except having a 90-degree angle.
• Trigonometry often uses laws such as the Law of Sines and the Law of Cosines to solve these triangles.

Recognizing that a triangle is oblique allows us to apply these specific properties to solve problems, especially in contexts where a right angle is absent.

Right Angle

A right angle is an angle of exactly 90 degrees. In the context of triangles, a right triangle features one right angle. The side opposite the right angle is called the hypotenuse, which is the longest side in a right triangle.

Here’s what you need to know about right angles and right triangles:

• They feature one 90-degree angle.
• The other two angles are acute and they sum up to 90 degrees.
• Commonly solved using the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse.

It is essential to identify these angles and their properties when solving problems related to right triangles, especially in practical fields like engineering and architecture.