Problem 7 - Algebra and Trigonometry

Question

If two angles of a triangle measure \(48^{\circ}\) and \(93^{\circ},\) what is the measure of the third angle? (a) \(132^{\circ}\) (b) \(77^{\circ}\) (c) \(42^{\circ}\) (d) \(39^{\circ}\)

Step-by-Step Solution

Answer

The measure of the third angle is \(39^{\text{°}}\).

Step 1 - Understanding the Triangle Angle Sum Property

The sum of the internal angles of any triangle is always equal to \(180^{\text{°}}\). This fundamental property will be used to find the measure of the missing angle.

Step 2 - Add the Known Angles

Given the angles are \(48^{\text{°}}\) and \(93^{\text{°}}\). Add these two angles together: \(48^{\text{°}} + 93^{\text{°}} = 141^{\text{°}}\).

Step 3 - Subtract from 180°

To find the measure of the third angle, subtract the sum of the known angles from \(180^{\text{°}}\): \(180^{\text{°}} - 141^{\text{°}} = 39^{\text{°}}\).

Step 4 - Verify the Solution

Verify the answer by adding all three angles to check if they sum up to \(180^{\text{°}}\): \(48^{\text{°}} + 93^{\text{°}} + 39^{\text{°}} = 180^{\text{°}}\). The calculation is correct.

Key Concepts

Internal Angles of a Triangle

In any triangle, the internal angles are the angles inside the triangle that are formed between two sides. Regardless of the shape or size of the triangle, there is a crucial property about these angles. The sum of the internal angles in a triangle is always equal to \(180^{\circ}\). This is known as the Triangle Angle Sum Property.
You can always remember this property because a straight line also measures \(180^{\circ}\), and if you think about slicing a triangle from vertex to base, it looks like you are stretching it into a line.

For example, in a triangle with angles \(48^{\circ}\) and \(93^{\circ}\), to find the third angle, simply subtract the sum of the given angles from \(180^{\circ}\). This property is helpful for solving many problems related to triangle measurements.

Triangular Geometry

Triangular geometry involves studying the properties and relations of triangles. Knowing the basic properties, like the Triangle Angle Sum Property, helps in understanding and solving more complex problems.

Triangles can be classified based on their angles or sides into different types, such as:

• Equilateral: All sides and internal angles are equal.
• Isosceles: Two sides and two internal angles are equal.
• Scalene: All sides and internal angles are different.

Also, based on their angles, you have:

• Acute: All three internal angles are less than \(90^{\circ}\).
• Right: One internal angle is exactly \(90^{\circ}\).
• Obtuse: One internal angle is greater than \(90^{\circ}\).

Understanding these classifications make it easier to solve problems like finding missing angles or sides of a triangle.

Angle Calculation

Calculating angles in a triangle often starts with known internal angles or sides. The primary steps include:

• Identify the known angles and sides.
• Use the Triangle Angle Sum Property for angles.
• Apply angle subtraction or addition.

For instance, if you are given triangles with angles \(48^{\circ}\) and \(93^{\circ}\), add these two to get \(141^{\circ}\). Then, subtract this from \(180^{\circ}\) to find the missing angle:
\[180^{\circ} - 141^{\circ} = 39^{\circ}\]\
Always double-check your calculation by adding all three angles to ensure they sum up to \(180^{\circ}\). This verification step ensures the solution's accuracy. Knowing and practicing these steps will bolster your problem-solving skills in geometry.