Problem 19 - Algebra A Combined Function

Question

The flag of Equatorial Guinea contains an isosceles triangle. (Recall that an isosceles triangle contains two angles with the same measure.) If the measure of the third angle of the triangle is \(30^{\circ}\) more than twice the measure of either of the other two angles, find the measure of each angle of the triangle. (Hint: Recall that the sum of the measures of the angles of a triangle is \(180^{\circ} .\) )

Step-by-Step Solution

Answer

The two equal angles are each \(37.5^{\circ}\) and the third angle is \(105^{\circ}\).

Step 1: Define variables for the angles

Let's define the measures of the unequal angles. Let \( x \) be the measure of the two equal angles in degrees. The measure of the third angle is given as \( 30^{\circ} \) more than twice \( x \). Thus, the third angle would be \( 2x + 30^{\circ} \).

Step 2: Set up the equation using angle sum property

We know that the sum of the angles in a triangle is always \( 180^{\circ} \). Therefore, we can set up the equation: \( x + x + (2x + 30^{\circ}) = 180^{\circ} \). Simplifying, this equation becomes \( 4x + 30^{\circ} = 180^{\circ} \).

Step 3: Solve for \( x \)

Subtract \( 30^{\circ} \) from both sides to isolate the terms with \( x \): \( 4x = 180^{\circ} - 30^{\circ} = 150^{\circ} \). Now, divide both sides by 4 to find \( x \): \( x = \frac{150^{\circ}}{4} = 37.5^{\circ} \).

Step 4: Calculate the measure of the third angle

Knowing that \( x = 37.5^{\circ} \), we substitute back into the expression for the third angle: \( 2x + 30^{\circ} = 2(37.5^{\circ}) + 30^{\circ} = 75^{\circ} + 30^{\circ} = 105^{\circ} \).

Key Concepts

Angle Measurement

In mathematics, an angle is a measure of the rotation or bend between two intersecting lines or rays. To find out the measurement of an angle, we measure how much one line has been rotated around the other line. This measurement is commonly given in degrees, with a full circle consisting of 360 degrees. Each degree is further divided into 60 minutes, and each minute is divided into 60 seconds.

When working with angles in a triangle, such as in the isosceles triangle mentioned in the problem, it's important to understand that we are usually dealing with angles in degrees. Each angle has a specific location and role within the shape. Careful measurement and calculation help ensure that we understand the relationships between these angles, which is essential to accurately solving geometric problems involving triangles.

Angle Properties

The properties of angles are essential in the study of geometrical shapes, especially triangles. In the case of an isosceles triangle, these properties have unique attributes that aid in solving problems. Here are some key properties:

• **Equal Angles:** In an isosceles triangle, two of its opposite angles are always the same.
• **Base Angles:** The angles adjacent to the equal sides are called base angles and are always congruent.
• **Vertex Angle:** The angle between the two equal sides differs in size from the base angles.

These properties help predict the triangle's configuration, aiding in solving for unknown angles. For example, if the vertex angle increases, the equal base angles must decrease to maintain the angle sum of the triangle. Similarly, if you are given perimeter details, the angle properties equip you to find out missing angles accurately.

Triangle Sum Theorem

The Triangle Sum Theorem is one of the fundamental principles that guides us in understanding triangles. This theorem states that the sum of all interior angles in any triangle is always 180 degrees. This principle is incredibly valuable when solving geometric problems and acts as a foundational rule for triangle calculations.

This theorem can be expressed in a simple mathematical equation: If a triangle has angles \( A \), \( B \), and \( C \), then \( A + B + C = 180^{\circ} \). During problem-solving, this theorem gives you a reliable technique to check if your calculated angles are correct. For instance, if you know two angles, you can easily find the third by subtracting the known angles from 180 degrees. Similarly, if you know one angle is missing, the theorem allows you to set up an equation to solve for that unknown angle, making it an essential tool in geometry.