Problem 74 - Algebra A Combined Function

Question

Dr. Dorothy Smith gave the students in her geometry class at the University of New Orleans the following question. Is it possible to construct a triangle such that the second angle of the triangle has a measure that is twice the measure of the first angle and the measure of the third angle is 5 times the measure of the first? If so, find the measure of each angle. (Hint: Recall that the sum of the measures of the angles of a triangle is \(180^{\circ} .\) )

Step-by-Step Solution

Answer

Yes, the triangle is possible with angles of 22.5°, 45°, and 112.5°.

Step 1: Define the Variables

Let the measure of the first angle be \( x \). According to the problem, the second angle is twice the first angle, so it can be represented as \( 2x \). Similarly, the third angle is five times the first angle, so it is \( 5x \).

Step 2: Set Up the Equation

The sum of the measures of the angles in a triangle must be \( 180^{\circ} \). Therefore, we can set up the equation: \( x + 2x + 5x = 180 \). This equation represents the sum of the three angles in terms of \( x \).

Step 3: Solve the Equation

Simplify the equation: \( x + 2x + 5x = 180 \) becomes \( 8x = 180 \). Solve for \( x \) by dividing both sides of the equation by 8: \( x = \frac{180}{8} = 22.5 \) degrees.

Step 4: Calculate Each Angle

Using \( x = 22.5 \):- The first angle is \( x = 22.5^{\circ} \).- The second angle is \( 2x = 2 \times 22.5 = 45^{\circ} \).- The third angle is \( 5x = 5 \times 22.5 = 112.5^{\circ} \).

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It's like the study of the world around us, where everything has a shape and form - from tiny atoms to towering buildings. When we think of geometry, we often imagine shapes like triangles, squares, and circles. These shapes help explain how our three-dimensional world is structured.
Triangles are especially important in geometry because they are the simplest polygon, with the fewest sides and angles. Understanding the properties of triangles helps us solve various mathematical problems and understand the space better. In geometry, you'll often find yourself working with points, lines, angles, and surfaces. It's the foundation of many advanced concepts in mathematics and science.

Exploring Angle Measures

Angles are everywhere. An angle is formed when two lines meet at a point, and it's measured in degrees. In mathematics, angles are crucial for understanding shapes and designs. When dealing with triangles, it's essential to remember that the sum of all interior angles in a triangle is \(180^{\circ}\). This fundamental rule helps solve problems like the one in the exercise given.
To explore angle measures, we divide a circle into 360 degrees – a full rotation. Simple geometry often involves smaller angles, such as those in triangles. When working with angles, we label them in letters or variables (like \(x \) for unknown angles) to make mathematical calculations easier. Always start by determining all known angles and what you're solving for. Use the angle sum theorem as your guide when working with triangles.

Triangle Properties Unveiled

Triangles have unique properties that make them both simple and complex. Each triangle has three sides and three angles, and these angles always add up to \(180^{\circ} \). This property is known as the Triangle Angle Sum Theorem, and it's a powerful tool in geometry. Beyond just their angles, triangles can be classified based on side length or angle measures:

• **Equilateral triangles** - where all sides and angles are equal. Each angle measures \(60^{\circ} \).
• **Isosceles triangles** - two sides are equal, and the angles opposite these sides are equal.
• **Scalene triangles** - all sides and angles are different.

Understanding triangle properties allows us to predict and calculate various aspects of their form. For example, if you know two angle measures, you can use the triangle angle sum theorem to find the third angle. Additionally, the relationships between the sides and angles can solve problems involving geometry in daily life, architecture, engineering, and more. When tackling challenging problems, break down the properties step-by-step to grasp the solution fully.