Problem 62 - Algebra A Combined Function

Question

The measures of the angles of a particular triangle are such that the second and third angles are each four times the measure of the smallest angle. Find the measures of the angles of this triangle.

Step-by-Step Solution

Answer

The angles of the triangle are 20°, 80°, and 80°.

Step 1: Define Variables

Let's denote the smallest angle by \( x \). According to the problem, the second and third angles are each four times the smallest angle. Therefore, the second and third angles are both \( 4x \).

Step 2: Set Up Equation

In a triangle, the sum of the angles is always \( 180^\circ \). We can set up the equation for the sum of the angles: \( x + 4x + 4x = 180 \).

Step 3: Simplify the Equation

Combine like terms in the equation \( x + 4x + 4x = 180 \) to get \( 9x = 180 \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation \( 9x = 180 \) by 9: \( x = \frac{180}{9} \). This simplifies to \( x = 20 \).

Step 5: Calculate Other Angles

Using the value of \( x \), calculate the other two angles: \( 4x = 4 \times 20 = 80 \). So, the second and third angles are both \( 80^\circ \).

Step 6: Verify the Solution

Check the sum of the angles: \( 20 + 80 + 80 = 180 \). This confirms that the angles satisfy the triangle angle sum property.

Key Concepts

Solving Equations in Geometry

When you come across problems in geometry, such as finding unknown angles in a triangle, the process often involves solving equations. Solving equations is a fundamental skill in mathematics used to find the value of unknown variables. In the exercise, the variable was labeled as \( x \) to represent the smallest angle. The other angles, being four times larger, were represented as \( 4x \).

Step-by-Step Solving:

• Define the variables based on given information.
• Set up an equation that represents the sum of the elements involved, like the angle sum of a triangle, which is \( 180^\circ \).
• Simplify the equation by combining like terms.
• Solve for the unknown value by isolating the variable, using basic arithmetic operations.

Breaking it down into these steps makes it manageable and systematic, turning complex problems into straightforward tasks. The main goal is to express relationships and find unknowns efficiently.

Understanding Angle Measures

An understanding of angle measures is crucial, especially when dealing with triangles. In geometry, an angle is typically measured in degrees, and for any triangle, the sum of the three interior angles is always \( 180^\circ \). This property is known as the triangle angle sum property, and it's one of the most important rules in geometry.

When tackling any triangle problem, the key is to use the information given to express unknown angles in terms of known measurements. The smallest angle in the exercise was denoted as \( x \), paving the way to express other angles relative to it, i.e., \( 4x \) in this case for the other two angles. This understanding not only aids in solving the problem but ensures that solutions are logical and consistent with geometric properties.

Exploring Triangle Properties

Triangles are simple, yet profoundly important shapes in geometry. They have unique properties, like the one highlighted in the exercise where the interior angles always sum up to \( 180^\circ \). Understanding these properties helps in various applications, from solving problems to designing structures.

Key Triangle Properties:

• Sum of interior angles is always \( 180^\circ \).
• Classification by angles: acute (all angles less than \( 90^\circ \)), right (one \( 90^\circ \) angle), obtuse (one angle more than \( 90^\circ \)).
• Classification by sides: equilateral (all sides equal), isosceles (two sides equal), scalene (all sides different).

Using the given properties, students can branch into understanding deeper concepts in geometry and tackle even more intricate problems with confidence. Recognizing these characteristics makes it easier to deconstruct problems and approach solutions logically.