Problem 27 - Prealgebra

Question

The smallest angle in a triangle is \(\frac{1}{3}\) as large as the largest angle. The third angle is twice the smallest angle. Find the three angles.

Step-by-Step Solution

Answer

The three angles are 30°, 60°, and 90°.

Step 1: Define Variables

Let's define the variables for the three angles. Suppose the smallest angle is \( x \), the largest angle is \( 3x \) (since it is \( \frac{1}{3} \) of the largest), and the third angle is \( 2x \).

Step 2: Use Triangle Angle Sum Property

The sum of angles in a triangle is always \( 180^\circ \). Therefore, the equation is: \[ x + 3x + 2x = 180 \] This simplifies to:\[ 6x = 180 \]

Step 3: Solve for Smallest Angle

Solve for \( x \) by dividing both sides of the equation by 6:\[ x = \frac{180}{6} = 30 \] Thus, the smallest angle is \( 30^\circ \).

Step 4: Find Largest Angle

The largest angle is \( 3x \), so:\[ 3x = 3 \times 30 = 90 \] Therefore, the largest angle is \( 90^\circ \).

Step 5: Find Third Angle

The third angle is twice the smallest angle, so:\[ 2x = 2 \times 30 = 60 \] Thus, the third angle is \( 60^\circ \).

Step 6: Conclusion

The three angles in the triangle are \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \). These satisfy the properties given in the problem.

Key Concepts

Understanding Angle Relationships

Angle relationships are foundational to solving many geometry problems, especially in triangles. In the triangle angle sum problem, we're looking at how the varying sizes of angles connect and relate to each other. Here, each angle is expressed in terms of a base angle, making them easy to compare and calculate.
A helpful way to understand angle relationships is to use a simple labeling system. In any triangle, you might encounter:

• Complementary angles, where the sum of two angles is 90°.
• Supplementary angles, where they sum up to 180°.
• The concept of interior angles of a triangle, which together always sum to 180°.

In our problem, one angle is described as a fraction or multiple of another. This familiar relationship lets us set up an equation easily by aligning every angle to a single variable base. It neatly ties into concepts like parallel line angles and congruent angles in different contexts, underscoring how useful understanding these relationships can be across geometry.

Solving Equations with Angles

Once we've established our variables, solving equations becomes a vital part of resolving the scenario. Here, we looked at an equation rooted in the triangle angle sum property, which is always 180°. Solving such equations requires us to:

• Identify what each angle represents in terms of a common variable, say \( x \).
• Express other angles as multiples or fractions of \( x \), as seen with \( 3x \) and \( 2x \) in the problem.
• Substitute these expressions into the equation reflecting the angle sum, yielding \( x + 3x + 2x = 180 \).

From here, we simplified the equation to \( 6x = 180 \), solved easily by dividing both sides by 6. This kind of algebraic process is fundamental because it applies mathematically robust approaches to solve geometry problems effectively, enhancing problem-solving fluency in more complex geometrical applications.

Navigating Geometry Problems

Tackling geometry problems goes beyond just crunching numbers. It’s about visualizing and understanding spatial relations and properties that guide how shapes function within given constraints.
Geometry problems, like the one with triangle angles, require tight connections between spatial reasoning and arithmetic skills—a seamless blend where:

• Visualizing the triangle and labeling its angles aligns with defining variables.
• Recognizing properties such as the angle sum rule, enhances comprehension.
• Flexibly transforming real-world constraints into mathematical equations enables systemic solutions.

This triangle problem particularly illustrates how insightful reasoning paves the way for uncovering solutions. Understanding how different geometric rules interact empowers you to translate a range of spatial problems into manageable mathematical tasks. Always consider revisiting familiar shapes and their properties—they’re often interlinked with many solutions across the math spectrum.