Problem 226 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. One angle of a right triangle measures 36.5 degrees. What is the measure of the other small angle?

Step-by-Step Solution

Answer

The measure of the other small angle is 53.5 degrees.

Step 1 - Understand the Properties of a Right Triangle

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

Step 2 - Identify the Given Information

One of the angles in the right triangle is given as 36.5 degrees. The right angle measures 90 degrees.

Step 3 - Set Up the Equation

Let the measure of the unknown angle be denoted as \( x \). Using the property that the sum of angles in a triangle is 180 degrees, we can write the equation: \[ 90^\text{o} + 36.5^\text{o} + x = 180^\text{o} \]

Step 4 - Solve for the Unknown Angle

To find the measure of the unknown angle, solve the equation: \[ x = 180^\text{o} - 90^\text{o} - 36.5^\text{o} \]Simplify the calculation: \[ x = 53.5^\text{o} \]

Key Concepts

angle sum of a triangle

The sum of the angles in any triangle is always 180 degrees. This is a fundamental property that applies to all triangles, including right triangles.

In our example, we have one right angle (90 degrees) and another angle that measures 36.5 degrees.

The formula to remember here is simple:
\( Angle_1 + Angle_2 + Angle_3 = 180° \)

Inserting the known values:
\( 90° + 36.5° + x = 180° \)

Solving for the unknown angle gives us:
\( x = 180° - 90° - 36.5° = 53.5° \)

So, the unknown angle measures 53.5 degrees.

right triangle

A right triangle has one angle equal to 90 degrees. This is known as the right angle. The other two angles in a right triangle must add up to 90 degrees because the total sum of angles in any triangle is 180 degrees.

Here's a quick summary of right triangle properties:

• One angle is always 90 degrees.
• The sum of the other two angles is always 90 degrees.
• It follows the Pythagorean theorem, although that isn't directly relevant to solving for angles.

For example:

\( Angle_1 = 90° (right angle) \)
\( Angle_2 = 36.5° (given) \)
\( Angle_3 = x (unknown) \)

Since we know the sum of Angle_2 and Angle_3 must be 90 degrees:
\( 36.5° + x = 90° \)
\( x = 90° - 36.5° \)
Therefore,
\( x = 53.5° \)

solving for unknown angles

Solving for unknown angles in a triangle involves understanding the properties and relationships between the angles.

Follow these steps to solve for an unknown angle in any triangle:

• Identify all given angles.
• Remember that the sum of all angles in any triangle is 180 degrees.
• If dealing with a right triangle, remember that one angle is 90 degrees.
• Set up an equation using the known angle values and the sum property.
• Solve for the unknown angle.

In our specific exercise:

• We were given one angle (36.5 degrees).
• We knew one angle was 90 degrees (right angle).
• We set up the equation: \( 90° + 36.5° + x = 180° \)
• We solved for x: \( x = 180° - 90° - 36.5° = 53.5° \)

The unknown angle in the triangle is therefore 53.5 degrees.