Problem 58 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. In \(\triangle A B C\), angle \(B\) is \(8^{\circ}\) less than one-half of angle \(A\) and angle \(C\) is \(28^{\circ}\) larger than angle \(A\). Find the measures of the three angles of the triangle.

Step-by-Step Solution

Answer

The angles are \(64^{\circ}\), \(24^{\circ}\), and \(92^{\circ}\).

Step 1: Set Up the Equations

Let the measure of angle \(A\) be \(x\) degrees. According to the problem, angle \(B\) is \(8^{\circ}\) less than half of angle \(A\), so angle \(B\) can be expressed as \(\frac{x}{2} - 8\). Additionally, angle \(C\) is \(28^{\circ}\) larger than angle \(A\), hence \(C = x + 28\).

Step 2: Use the Triangle Angle Sum Property

The sum of the measures of the angles in a triangle is always \(180^{\circ}\). Therefore, we can write the equation: \[ x + \left(\frac{x}{2} - 8\right) + (x + 28) = 180. \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ x + \frac{x}{2} - 8 + x + 28 = 180. \] This simplifies to \[ 2.5x + 20 = 180. \]

Step 4: Solve for \(x\)

Subtract \(20\) from both sides to get: \[ 2.5x = 160. \] Then, divide both sides by \(2.5\) to find \(x\): \[ x = 64. \] So, angle \(A\) measures \(64^{\circ}\).

Step 5: Solve for Angle B

Substitute \(x = 64\) back into the expression for angle \(B\): \[ B = \frac{64}{2} - 8 = 32 - 8 = 24^{\circ}. \]

Step 6: Solve for Angle C

Substitute \(x = 64\) back into the expression for angle \(C\): \[ C = 64 + 28 = 92^{\circ}. \]

Step 7: Verify the Solution

Check that the sum of the measures of the angles is \(180^{\circ}\): \[ 64 + 24 + 92 = 180. \] Since this equation holds true, the solution is verified.

Key Concepts

Triangle Angle Sum

In every triangle, no matter its type or size, the sum of its angles will always be exactly 180 degrees. This is known as the "triangle angle sum" property and is a fundamental rule in geometry. Understanding this concept is crucial for solving problems related to triangle angles.

When given certain expressions or unknowns for the angles, the triangle angle sum property allows us to construct an equation. This is essential in calculating the measures of the unknown angles when combined with algebraic expressions.

• All triangle types (equilateral, isosceles, scalene) adhere to this rule.
• To use this property, simply add up all provided expressions for the angles of the triangle and equate them to 180 degrees.

Through this principle, you can determine the measure of unknown angles in any triangle situation. The consistent total always totals up to the same value, making calculations predictable and reliable.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two expressions. They often contain variables that we need to solve for, such as in our problem with triangle angles. By setting up a correct algebraic equation, we can determine unknown values efficiently.

In our given exercise, each angle of the triangle was expressed in relation to the others using variables and constants.

• Angle A was designated as variable \(x\), the main unknown to find first.
• Angle B was expressed algebraically as \(\frac{x}{2} - 8\), showing its dependency on angle A.
• Angle C was calculated as \(x + 28\), another expression relying on the variable \(x\).

These algebraic expressions help us find the measures by substituting and solving these expressions based on the known sum of triangle angles (180 degrees). Solving algebraic equations involves combining like terms, isolating the variable, and performing arithmetic operations to deduce the measures of angle A, B, and C in sequence.

Angle Relationships

Understanding the relationships between angles is essential in solving triangle problems, particularly when angles are interdependent. Relationships like supplementary (adding up to 180 degrees), complementary (adding up to 90 degrees), or in this case, how one angle can be expressed in terms of another, are important.

In our problem, the angles are linked through specific relationships:

• Angle B is explicitly described as being \(8\) degrees less than half of angle A.
• Angle C is described as \(28\) degrees greater than angle A.

These relationships are translated into algebraic expressions, which allow us to solve for each angle systematically using the information provided. Understanding how to translate narrative relationships into mathematical expressions is key to solving these problems. Clearly mapping out these relationships is important to avoid confusion and ensures that the triangular angle sum is satisfied once solved.