Problem 224 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. One angle of a right triangle measures 51 degrees. What is the measure of the other small angle?

Step-by-Step Solution

Answer

The measure of the other small angle is 39 degrees.

Step 1: Understand Triangle Properties

A right triangle has one angle that is always 90 degrees. The remaining two angles must add up to 90 degrees because the sum of all angles in any triangle is 180 degrees.

Step 2: Identify Given Information

We are given that one of the smaller angles in the right triangle is 51 degrees.

Step 3: Subtract to Find the Unknown Angle

To find the measure of the other small angle, subtract the given angle from 90 degrees. Use the equation: \[ \text{Other Angle} = 90^\text{°} - 51^\text{°} \]

Step 4: Calculate the Measure

Subtract 51 from 90: \[ \text{Other Angle} = 90^\text{°} - 51^\text{°} = 39^\text{°} \]

Key Concepts

Triangle Properties

Triangles are fascinating shapes with unique properties. One key property of a triangle is that the sum of its internal angles is always 180 degrees. This holds true for all types of triangles, including right triangles. By understanding this property, we can easily solve various geometry problems.

• A right triangle always has one angle that is exactly 90 degrees.
• The other two angles in a right triangle must sum up to 90 degrees as well.
• This is because 180 degrees (total for any triangle) minus the 90 degrees (right angle) leaves 90 degrees for the other two angles.

These properties are not just theoretical; they help you solve real-world problems, like finding the measure of unknown angles in right triangles.

Angle Sum

Understanding the sum of angles is crucial in geometry. As previously mentioned, the internal angles of a triangle always add up to 180 degrees. This concept is especially useful in solving problems related to right triangles.
In right triangles, when we know one of the non-right angles, we can find the other by utilizing the angle sum property. Given one small angle, we can simply subtract its measure from the total of 90 degrees (since one angle is already 90 degrees).

• This property helps quickly determine missing angle measures.
• It is an essential concept for further studies in geometry and trigonometry.

Practicing with various right triangle problems will enhance your understanding of this fundamental concept.

Subtraction in Geometry

Geometry often requires the subtraction of known angle measures to find unknown ones. This straightforward method is highly effective in problems involving right triangles, as demonstrated in the original exercise.
In the exercise, we used the following steps to find the unknown angle:

• We identified that one angle is 51 degrees.
• We used the right triangle property where the sum of the two smaller angles must be 90 degrees.
• We subtracted the given angle from 90 degrees using the equation: \[ \text{Other Angle} = 90^\text{°} - 51^\text{°} \]
• Finally, we performed the calculation to get: \[ \text{Other Angle} = 39^\text{°} \]

This process shows the importance of basic arithmetic in solving geometry problems. Subtraction helps us navigate through complex problems by breaking them down into simpler steps.