Problem 61 - Algebra 2

Question

ACT/SAT The measure of the largest angle of a triangle is 14 less than twice the measure of the smallest angle. The third angle is 2 more than the measure of the smallest angle. What is the measure of the smallest angle? \(\begin{array}{rllllll}{\mathbf{F}} & {46} & {\mathbf{G} 48} & {\mathrm{H} 50} & {\mathbf{J}} & {82}\end{array}\)

Step-by-Step Solution

Answer

The smallest angle measures 48 degrees (option G).

Step 1: Understand the Triangle Angle Sum Property

In any triangle, the sum of the internal angles is always 180 degrees. This property will be useful for setting up an equation to solve the problem.

Step 2: Set up Variables for Angles

Let \( x \) represent the measure of the smallest angle. Therefore, according to the problem, the largest angle is \( 2x - 14 \) and the third angle is \( x + 2 \).

Step 3: Formulate the Equation

Using the triangle angle sum property, add up the angles: \( x + (2x - 14) + (x + 2) = 180 \). Simplify the equation to solve for \( x \).

Step 4: Simplify and Solve the Equation

First, simplify the equation to: \( 4x - 12 = 180 \). Then, solve for \( x \) by adding 12 to both sides: \( 4x = 192 \). Next, divide both sides by 4 to find \( x \): \( x = 48 \).

Step 5: Verify the Solution

Substitute \( x = 48 \) back into the expressions for the largest and third angles to verify: - Largest angle: \((2 \times 48) - 14 = 82\) - Third angle: \(48 + 2 = 50\)Add these angles: \(48 + 82 + 50 = 180\). The sum confirms the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that allows us to find unknown values. In this problem, the unknown we want to determine is the measure of the smallest angle of a triangle. We start by assigning a variable, typically represented as \( x \), to this unknown. Given the relationships between angles provided in the problem, we translate these verbal expressions into mathematical expressions. For instance, the problem states that the largest angle is 14 less than twice the smallest angle, which we represent as \( 2x - 14 \). The third angle is described as 2 more than the smallest angle, thus \( x + 2 \).

Once these expressions are set up, we apply the triangle angle sum property to form an equation. This sum is crucial as it lets us construct the equation \( x + (2x - 14) + (x + 2) = 180 \). By simplifying this equation, we gather like terms and isolate \( x \). Solving the linear equation by performing operations such as addition, subtraction, and division allows us to find the value of \( x \). This step-by-step approach helps ensure accuracy and clarity in solving for unknowns.

Triangle Geometry

Triangle geometry explores the properties and measures of triangles' sides and angles. An important concept within triangle geometry is the Triangle Angle Sum Property, which dictates that the sum of the interior angles of any triangle is always 180 degrees. This foundational rule is a critical tool for solving various problems involving triangles.

In our exercise, this property is applied to establish the equation that sums the angles of the triangle to 180 degrees. Understanding this constant sum allows us to decompose the problem, utilizing given relationships between angles to set up the relevant expressions. This process demonstrates how geometry provides a framework for systematically approaching and solving real-world and theoretical problems.

To further explore triangle geometry, one might delve into different types of triangles such as equilateral, isosceles, and scalene, each having unique properties and angle relationships. Recognizing these types in problems can offer additional insights or simplifications when solving them.

ACT/SAT Math Problems

Math problems encountered in the ACT/SAT exams often integrate multiple mathematical concepts, requiring a blend of skills such as algebra, geometry, and logical reasoning. The question of finding the smallest angle of a triangle combines these topics by using equation-solving techniques and geometric properties. Students often face problems requiring similar methods of translating word descriptions into mathematical equations, a key skill evaluated in standardized tests.

To tackle these problems successfully, practice and familiarity with common problem types are imperative. Recognizing patterns, such as the use of the Triangle Angle Sum Property, and understanding how to express relationships algebraically, such as with the angle expressions given, forms an essential part of the preparation. Strategies like practicing the simplification of equations and verifying solutions tightens accuracy.

Additionally, mastering time management and efficient problem-solving techniques can greatly enhance performance. Students are encouraged to practice turning word problems into equations efficiently and to verify answers for correctness, ensuring all parts of a problem are addressed.