Problem 214 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. The measures of two angles of a triangle are 47 and 72 degrees. Find the measure of the third angle.

Step-by-Step Solution

Answer

The measure of the third angle is 61 degrees.

Step 1 - Identify the Properties of a Triangle

Recall that the sum of the angles in any triangle is always 180 degrees.

Step 2 - Add the Known Angles

Add the measures of the two known angles: 47 degrees and 72 degrees. \[ 47^\text{°} + 72^\text{°} = 119^\text{°} \]

Step 3 - Subtract from 180 Degrees

Subtract the sum of the known angles from 180 degrees to find the measure of the third angle. \[ 180^\text{°} - 119^\text{°} = 61^\text{°} \]

Key Concepts

Angle Calculation

When solving problems involving triangles, knowing how to calculate angles is essential. Here, we are given two angles of a triangle, and we need to find the third angle.
The sum of the measures of the angles in a triangle is always 180 degrees. This is a fundamental rule in geometry. By knowing two angles, we can use basic arithmetic steps to find the missing angle.
You first add the measures of the known angles. In our exercise, we have 47 degrees and 72 degrees. Adding these, we get: \[ 47^\text{°} + 72^\text{°} = 119^\text{°} \]
Next, subtract the sum from 180 degrees to find the measure of the third angle. Thus, the third angle is: \[ 180^\text{°} - 119^\text{°} = 61^\text{°} \]
So, the measure of the third angle is 61 degrees.

Triangle Properties

Understanding the properties of triangles is crucial in geometry. A triangle has three sides and three angles.
Some key properties include:

• The sum of the angles in a triangle is always 180 degrees.
• The length of any side must be less than the sum of the other two sides but greater than their difference.
• Triangles can be classified by their sides (equilateral, isosceles, and scalene) or by their angles (acute, right, and obtuse).

Knowing these properties helps solve various problems involving triangles. For our particular exercise, using the property that the angles sum up to 180 degrees enables us to find the missing angle.

Basic Geometry

Basic geometry involves the study of points, lines, shapes, and their properties. Triangles are fundamental shapes studied in geometry because of their simplicity and wide application.
Key geometric concepts related to triangles include:

• Angles and their measures.
• Properties of different types of triangles.
• Relationships between sides and angles.

In the exercise, basic geometry principles help us understand that all angles in a triangle sum up to 180 degrees. This foundational concept allows for the simple calculation of a missing angle when the other two are known.

Angle Sum Property

One of the most important properties of a triangle is the angle sum property. This states that the sum of the interior angles of a triangle is always 180 degrees.
This property is essential for solving many geometric problems, including finding unknown angles within a triangle.
Here’s an application: given two angles of a triangle, \[ 47^\text{°} \text{ and } 72^\text{°} \] we simply add these together \[ 47^\text{°} + 72^\text{°} = 119^\text{°} \]
Next, we use the angle sum property to find the third angle by subtracting this sum from 180 degrees: \[ 180^\text{°} - 119^\text{°} = 61^\text{°} \]
Therefore, the measure of the third angle is 61 degrees, showing the practicality and simplicity of this property.

Elementary Algebra

Elementary algebra often comes into play in geometry, especially in solving for unknowns. In our assignment, we used basic algebraic operations like addition and subtraction.
Steps involved include:

• Identify known values (angles given as 47 and 72 degrees).
• Add the known angles together: \[ 47^\text{°} + 72^\text{°} = 119^\text{°} \]
• Subtract the sum from 180 degrees to find the unknown angle: \[ 180^\text{°} - 119^\text{°} = 61^\text{°} \]

This process is straightforward and showcases how elementary algebra aids in solving geometric problems. By using basic operations, we can easily navigate through problems involving triangle properties.