Problem 5 - Elementary and Intermediate Algebra

Question

Solve. Geometry. In triangle \(A B C,\) the measure of angle \(B\) is three times that of angle \(A .\) The measure of angle \(C\) is \(20^{\circ}\) more than that of angle \(A .\) Find the angle measures.

Step-by-Step Solution

Answer

Angle A: 32°, Angle B: 96°, Angle C: 52°.

Step 1 - Set Up the Relationships

Let the measure of angle A be represented by \(A = x\). According to the problem, the measure of angle B is three times that of angle A, so \(B = 3x\). The measure of angle C is 20° more than the measure of angle A, so \(C = x + 20^\circ\).

Step 2 - Use the Triangle Angle Sum Property

The sum of the angles in any triangle is \(180^\circ\). Therefore, the sum of angles A, B, and C is: \[ A + B + C = 180^\circ. \]

Step 3 - Substitute the Known Values

Substitute the expressions for each angle into the equation: \[ x + 3x + (x + 20^\circ) = 180^\circ \]

Step 4 - Simplify and Solve for x

Combine like terms and solve for x: \[ 5x + 20^\circ = 180^\circ \] \[ 5x = 160^\circ \] \[ x = 32^\circ \]

Step 5 - Determine All Angle Measures

With \(x = 32^\circ\), calculate the measures of angles A, B, and C: \[ A = x = 32^\circ \] \[ B = 3x = 3 \times 32^\circ = 96^\circ \] \[ C = x + 20^\circ = 32^\circ + 20^\circ = 52^\circ \]

Key Concepts

Angle Relationships

Understanding angle relationships helps in connecting different angles within a geometric shape, such as a triangle. In our problem, we establish the relationship between the angles by setting angle A equal to a variable, say, x. Angle B is three times angle A, thus B = 3x. Angle C is 20° more than angle A, so C = x + 20°. These relationships are crucial for solving problems involving multiple angles. By setting up these relationships, you can create a basis for forming equations and solving them to find the unknown angles.

Algebra in Geometry

Algebra helps in solving geometric problems by translating them into algebraic expressions and equations. In our triangle problem, we use algebra to express angles B and C in terms of x, where x represents angle A. The problem becomes finding the value of x that satisfies the sum of the triangle angles, which is always 180°. Here's the algebraic setup:

• A = x
• B = 3x
• C = x + 20°

The angle sum property of a triangle tells us that these angles must add up to 180°: x + 3x + (x + 20°) = 180°. Simplifying this, we get: 5x + 20° = 180°. Solving for x gives us x = 32°. Algebra is used here to solve for x and find actual measures of all angles in the triangle.

Triangle Properties

Triangles have some very important properties, one of which is the Triangle Angle Sum Property. This property states that the sum of the interior angles of a triangle is always 180°. This principle is foundational and applies to all types of triangles. Additionally, it's critical to know how to apply this property to find unknown angles. Using our relationships for angles A, B, and C, and their algebraic expressions, we set up the equation x + 3x + (x + 20°) = 180°, which simplifies to 5x + 20° = 180°. From here, we find x = 32°. With this value, we determine the measures of angles A, B, and C as 32°, 96°, and 52° respectively. Understanding triangle properties not only helps in solving these problems but also enhances basic geometric intuition and problem-solving skills.