Problem 17 - Intermediate Algebra

Question

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In \(\Delta A B C, \angle A\) measures \(100^{\circ}\) less than the sum of the measures of \(\angle B\) and \(\angle C,\) and the measure of \(\angle C\) is \(40^{\circ}\) less than twice the measure of \(\angle B .\) Find the measure of each angle of the triangle.

Step-by-Step Solution

Answer

Angle A is \(40^{\circ}\), angle B is \(60^{\circ}\), angle C is \(80^{\circ}\).

Step 1: Understand the problem

We need to find the measures of angles \(\angle A\), \(\angle B\), and \(\angle C\) in triangle \(\Delta ABC\). We know two key facts: The sum of all angles in a triangle equals \(180^{\circ}\), \(\angle A\) is \(100^{\circ}\) less than the sum of \(\angle B\) and \(\angle C\), and \(\angle C\) is \(40^{\circ}\) less than twice \(\angle B\).

Step 2: Express relationships as equations

Let the measures of angles \(B\) and \(C\) be \(b\) and \(c\) respectively. According to the problem, \(A = (b + c) - 100^{\circ}\) and \(c = 2b - 40^{\circ}\). Also, \(a + b + c = 180^{\circ}\). We'll use these equations to find each angle.

Step 3: Substitute and eliminate

Substitute \(c = 2b - 40^{\circ}\) into the equation \(a = b + c - 100^{\circ}\) to express \(a\) in terms of \(b\): \[ a = b + (2b - 40) - 100 = 3b - 140^{\circ} \].

Step 4: Form a single equation in one variable

Now substitute \(a = 3b - 140^{\circ}\) and \(c = 2b - 40^{\circ}\) into \(a + b + c = 180^{\circ}\): \[ (3b - 140) + b + (2b - 40) = 180 \] Simplify to form the equation: \[ 6b - 180 = 180 \].

Step 5: Solve for angle B

Add \(180\) to both sides to isolate \(b\): \[ 6b = 360 \] Divide both sides by 6 to find \(b\): \[ b = 60^{\circ} \].

Step 6: Find angle C using B's value

Substitute \(b = 60^{\circ}\) back into \(c = 2b - 40^{\circ}\): \[ c = 2(60) - 40 = 120 - 40 = 80^{\circ} \].

Step 7: Find angle A using B and C's values

Use the expression \(a = 3b - 140^{\circ}\) or \(a = b + c - 100^{\circ}\) with \(b = 60^{\circ}\) and \(c = 80^{\circ}\): \[ a = 60 + 80 - 100 = 40^{\circ} \].

Step 8: Verify the angle sum

Check if \(a + b + c = 180^{\circ}\): \[ 40 + 60 + 80 = 180 \]. This confirms the calculations and verifies the solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, particularly when analyzing geometric relationships. In order to determine unknown values, we translate the given conditions of a problem into mathematical equations.
In this exercise, we are tasked with finding the measures of the angles in triangle \(\Delta ABC\). Given the relationships between the angles, our first step is to express these conditions as equations.

• Equation 1: Reflects that \(\angle A\) is \(100^{\circ}\) less than the sum of \(\angle B\) and \(\angle C\): \(A = (b + c) - 100^{\circ}\).
• Equation 2: Tells us \(\angle C\) is \(40^{\circ}\) less than twice \(\angle B\): \(c = 2b - 40^{\circ}\).
• Equation 3: Represents the triangle angle sum property: \(a + b + c = 180^{\circ}\).

By solving these three equations simultaneously, using substitution and elimination, we eliminate variables step-by-step to find the measures of \(\angle A\), \(\angle B\), and \(\angle C\).

Triangle Properties

Triangles have unique properties that define their behavior and allow us to solve for unknowns. One such critical property is the Triangle Angle Sum Property. This property states that the sum of the interior angles of any triangle is always \(180^{\circ}\).
When dealing with triangle problems, understanding and leveraging these intrinsic properties is crucial:

• Angle Sum: The three interior angles \(a\), \(b\), and \(c\) of triangle \(\Delta ABC\) add up to \(180^{\circ}\).
• Types of Angles: While considering the type of triangle formed, knowing whether the triangle is acute, obtuse, or right can help predict what the angle measures might be.

Given the degrees of freedom and variables in triangle problems, these foundational properties help narrow down possible values and allow us to systematically approach the problem.

Angle Relationships

Understanding angle relationships enables us to link given conditions to mathematical expressions. In the context of a triangle, these relationships often involve comparisons between angles, which can provide insights into their absolute measures.
In our exercise, we have:

• \(\angle A\) is described in relation to the sum of \(\angle B\) and \(\angle C\). This provides a relational expression aiding in solving for \(\angle A\): \(A = (b + c) - 100^{\circ}\).
• \(\angle C\) is positioned relative to \(\angle B\), allowing the expression \(c = 2b - 40^{\circ}\).

These expressions are part of a system of equations that, when solved together, yield the exact measurements of the angles. Recognizing how these relationships manifest in the form of algebraic expressions equips us to tackle and solve such geometric problems with confidence.