Problem 212 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

Step-by-Step Solution

Answer

The measure of the third angle is 35 degrees.

Step 1: Recall the Triangle Angle Sum Property

The sum of the interior angles of a triangle is always 180 degrees.

Step 2: Set Up the Equation

Let the measure of the third angle be represented by the variable \( x \). We know that the sum of all angles in a triangle is 180 degrees. Therefore, the equation will be: \[ 61^\text{°} + 84^\text{°} + x = 180^\text{°} \]

Step 3: Solve for the Third Angle

Combine the measures of the known angles and solve for \( x \): \[ 61^\text{°} + 84^\text{°} = 145^\text{°} \] \[ 145^\text{°} + x = 180^\text{°} \] Subtract \( 145^\text{°} \) from both sides: \[ x = 180^\text{°} - 145^\text{°} \] \[ x = 35^\text{°} \]

Key Concepts

Interior Angles

When we talk about interior angles in a triangle, we mean the angles found inside the triangle, where the triangle's sides meet. A key fact to remember is that the sum of these interior angles is always 180 degrees. This property helps us solve for unknown angles.

For instance, if you know two angles of a triangle, you can find the third one easily. Just subtract the sum of the known angles from 180 degrees. This is very useful for solving many geometry problems involving triangles. Understanding this rule is one of the first steps in mastering geometry.

Triangle Properties

Triangles come with a set of properties that are important for solving many problems. One of the most fundamental properties is the Triangle Angle Sum Property. As mentioned earlier, the sum of all interior angles in any triangle is always 180 degrees.

There are different types of triangles, such as:

• Equilateral triangles, with all three angles equal to 60 degrees
• Isosceles triangles, where at least two angles are equal
• Scalene triangles, where all angles are different

To solve problems related to triangles, knowing these properties and types can be very helpful. For example, if you know a triangle has two angles equal, it’s an isosceles triangle, and you can easily find the missing angle.

When working with triangles, always remember these properties and rules. They will often guide you in the right direction.

Solving Equations

Understanding how to solve equations is crucial for finding unknown angles in triangles. Often, we set up an equation based on the properties we know. In this case, we use the fact that the sum of the interior angles in a triangle is 180 degrees.

Here’s a simple step-by-step approach:

• Identify what you need to find. For example, if you need the third angle, designate it as a variable, say, \( x \).
• Set up an equation including all known angles and the variable. For instance, if you know two angles are 61 degrees and 84 degrees, your equation would be: \( 61^\text{°} + 84^\text{°} + x = 180^\text{°} \)
• Simplify the equation by combining like terms. Here, you'd add 61 and 84 to get 145. So now, \( 145^\text{°} + x = 180^\text{°} \).
• Solve for the unknown by isolating the variable. This means subtracting 145 from 180: \( x = 180^\text{°} - 145^\text{°} \), to get \( x = 35^\text{°} \)

This systematic approach can be applied to many other problems involving geometry and algebra.

Practice solving these types of equations often so you get comfortable with the steps.