Problem 223 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

Step-by-Step Solution

Answer

57 degrees

Step 1: Understand the properties of a right triangle

In a right triangle, one of the angles is always 90 degrees. The sum of all angles in any triangle is 180 degrees.

Step 2: Determine the sum of the other two angles

The sum of the two remaining angles in a right triangle must be 90 degrees because the total sum of the angles is 180 degrees and one angle is already 90 degrees.

Step 3: Subtract the given angle from 90 degrees

Since one of the other angles is given as 33 degrees, subtract this from 90 degrees to find the measure of the other small angle: \[ 90^\text{°} - 33^\text{°} = 57^\text{°} \]

Key Concepts

Sum of Angles in a Triangle

Triangles are fundamental shapes in geometry. One key property of any triangle is that the sum of its internal angles is always 180 degrees. This means that if you know two angles in a triangle, you can always find the third one by subtracting the sum of the known angles from 180 degrees.

For instance, if a triangle has angles of 50 degrees and 60 degrees, the sum of these two angles is 110 degrees. The third angle would, therefore, be: \[180^\text{°} - 110^\text{°} = 70^\text{°} \]
This property is a key tool in solving many geometric problems, including those involving right triangles.

Right Triangle Angle Calculation

A right triangle has a special angle, which is always 90 degrees. Because of this, you only need to know one of the other two angles to find the third. The reason is simple: since the sum of all angles in any triangle is 180 degrees and you already have a 90-degree angle, the sum of the other two angles must be: \[180^\text{°} - 90^\text{°} = 90^\text{°} \]

So, if you know one of the other angles, you subtract it from 90 degrees to find the remaining angle. For example, in a right triangle where one of the angles measures 33 degrees, the other angle can be found by: \[90^\text{°} - 33^\text{°} = 57^\text{°} \]

This straightforward calculation helps quickly determine needed angles in various real-world problems, from construction to navigation.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. Triangles are one of the simplest and most studied shapes in geometry. They are classified based on their angles and sides:

• Equilateral triangles have all equal sides and angles of 60 degrees each.
• Isosceles triangles have at least two equal sides and two equal angles.
• Scalene triangles have all sides and angles different.

Yet, the right triangle is particularly significant. It's used in various applications, from engineering to art, due to its unique properties.

Understanding right triangles is essential since they form the basis of trigonometry, which involves studying relationships between angles and sides in triangles.