Problem 54 - Algebra 1

Question

In any triangle, the sum of the measures of the angles is \(180^{\circ}\). In triangle \(A B C, \angle A\) is four times as large as \(\angle B\). Angle \(C\) measures \(20^{\circ}\) less than \(\angle B\). Find the measure of each angle.

Step-by-Step Solution

Answer

\(\angle B \approx 33.33^{\circ}, \angle A \approx 133.33^{\circ}, \angle C \approx 13.34^{\circ}\)

Step 1 - Set Up the Equations

Let's denote \(\angle B\) as 'x'. According to the problem, \(\angle A\) is four times as large as \(\angle B\), so \(\angle A\) can be expressed as '4x'. Additionally, \(\angle C\) measures \(20^{\circ}\) less than \(\angle B\), so \(\angle C\) can be denoted as 'x - 20'. Now, using the rule that in any triangle, the sum of the angles equals \(180^{\circ}\), create the equation: \(4x + x + (x - 20) = 180\).

Step 2 - Simplify the Equation

First, combine like terms on the left side of the equation: \(6x - 20 = 180\). Then, add 20 to both sides of the equation: \(6x = 200\).

Step 3 - Solve for x

Next, divide both sides of the equation by 6 in order to isolate 'x', which will give \(x \approx 33.33^{\circ}\).

Step 4 - Find the measures of each angle

Substitute \(x \approx 33.33^{\circ}\) into the expressions for each angle. Thus, \(\angle B \approx 33.33^{\circ}\), \(\angle A = 4x \approx 133.34^{\circ}\), and \(\angle C = x - 20 \approx 13.33^{\circ}\). Remember that these values are approximate because we rounded the value of 'x'.

Key Concepts

Angle Measurement

Understanding angle measurement is essential in solving problems involving triangles. An angle is measured in degrees, with a full circle being 360 degrees. In a triangle, the sum of the measures of the three angles always adds up to 180 degrees. This is a fundamental property of triangles and is crucial for calculating unknown angles when some are given.

When measuring angles, we often use tools like protractors, but mathematics allows us to set up equations to find unknown angles using relationships and properties. By understanding how to assign values to these angles in terms of a variable, such as denoting one angle as 'x', we can effectively solve for all unknowns.

Angle Equations

Creating equations is a way to explore relationships between angles in a triangle. In the given problem, angle equations are set up based on the relationships between angles A, B, and C. You start by assigning a variable to one of the angles, such as \(\angle B\) as 'x'.

Next, we establish other angles in terms of this variable:

• \(\angle A = 4x\)
• \(\angle C = x - 20\)

These equations reflect how angles relate to each other, taking into account the specifications given in the exercise. Adding these angles together and equating them to 180 degrees allows us to solve for 'x'.

Once 'x' is found, substitute it back into the angles' equations to get the explicit measurements of each angle. This technique is widely used in geometry to solve for unknown angles efficiently.

Triangle Properties

Triangles hold specific properties that make them unique. One key property, as highlighted in this exercise, is that all internal angles of a triangle add up to 180 degrees. This applies to all triangles, whether they are equilateral, isosceles, or scalene.

Understanding these properties can help with solving problems like this one, where knowing the total sum of the angles allows us to calculate individual angles. Other essential properties worth exploring include:

• Equilateral triangles have three equal angles, each measuring 60 degrees.
• Isosceles triangles have two equal angles.
• Scalene triangles have no equal angles.

In this exercise, recognizing these properties helps in setting up the equations correctly and ensuring the solution works within the rules governing triangles.