Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank(s). A(n) _______ triangle has no right angles.

Step-by-Step Solution

Answer

The correct answer could either be 'obtuse triangle' or 'acute triangle'.

Step 1: Identify the kinds of triangle

There are several types of triangles including: an equilateral triangle (all sides and angles equal), an isosceles triangle (two sides and two angles equal), a scalene triangle (no sides or angles equal), a right triangle (one angle equals 90 degrees), an acute triangle (all angles are less than 90 degrees), and an obtuse triangle (one angle is more than 90 degrees).

Step 2: Matching definition

The question is looking for a type of triangle that does not contain a right angle. Looking at our list of different types of triangles, an obtuse triangle, an acute triangle and a scalene triangle could all potentially have no right angles.

Step 3: Correct Answer

The correct answer could either be an obtuse triangle or an acute triangle as these two types of triangle are defined as having no right angles.

Key Concepts

Acute Triangle

An acute triangle is a type of triangle where all three angles are less than 90 degrees. This means each angle is a sharp or "acute" angle, hence the name. This characteristic ensures that an acute triangle has no right angle, which makes it quite distinct from a right triangle. When working with an acute triangle, it's important to note:

• All angles are less than 90 degrees.
• The sum of all angles, like any triangle, is always 180 degrees.
• An acute triangle can be scalene, isosceles, or equilateral, depending on its side lengths.

For example, in an equilateral acute triangle, not only are all angles acute, but each angle is exactly 60 degrees. Remember, a triangle cannot be both right and acute as these two categories are mutually exclusive.

Obtuse Triangle

An obtuse triangle is characterized by having one angle that measures more than 90 degrees. This makes it recognizable as containing an "obtuse" angle. Like an acute triangle, an obtuse triangle will not have a right angle, keeping it away from the right-angle category. Key properties of an obtuse triangle include:

• One angle greater than 90 degrees (obtuse) and the other two angles are less than 90 degrees (acute).
• The sum of all angles remains 180 degrees.
• It often appears in scalene or isosceles form, but never equilateral since equilateral triangles have equal angles (all acute).

When identifying an obtuse triangle, look for that unique larger angle which sets it apart, as it creates an extended side that is visibly longer than the others.

Scalene Triangle

A scalene triangle is defined by having no equal sides and no equal angles. This lack of symmetry differentiates it from other triangle types like isosceles and equilateral triangles. Importantly, a scalene triangle could never possess a right angle by necessity, but it might or might not. Here are some essential characteristics:

• Each side of a scalene triangle is a different length.
• Each angle is a different measure.
• A scalene triangle can be acute or obtuse.

For instance, if all angles in a scalene triangle are less than 90 degrees, it's an acute scalene triangle. Conversely, if one angle is more than 90 degrees, it becomes an obtuse scalene triangle. Despite its irregularity, the scalene triangle abides by the fundamental rule that the sum of its angles is 180 degrees.