Problem 25 - Prealgebra

Question

Two angles in a triangle are equal, and their sum is equal to the third angle in the triangle. What are the measures of each of the three interior angles?

Step-by-Step Solution

Answer

The measures of the angles are 45°, 45°, and 90°.

Step 1: Understand the Problem

In a triangle with angles \( A \), \( B \), and \( C \), we have \( A = B \) and \( A + B = C \). Since \( A + B + C = 180° \) in any triangle, these conditions will help us find the individual angles.

Step 2: Express Given Conditions Mathematically

We express the conditions as equations: \( A = B \) and \( A + B = C \). Since \( A = B \), this implies \( 2A = C \). Also, \( A + B + C = 180° \).

Step 3: Simplify the Equations

Substitute \( C = 2A \) from the condition \( 2A = C \) into \( A + B + C = 180° \) to get: \( A + A + 2A = 180° \), which simplifies to \( 4A = 180° \).

Step 4: Solve for Angle A (and B)

Divide each side of \( 4A = 180° \) by 4 to find \( A \): \( A = 45° \). With \( A = B = 45° \) (since \( A = B \)), we have two of the angles.

Step 5: Solve for Angle C

Use the fact \( C = 2A \) to find \( C \): \( C = 2 imes 45° = 90° \).

Key Concepts

Interior Angles

Triangles have three interior angles, and the angles always lie inside the triangle itself. It is important to note that every triangle follows a basic rule where the sum of all three interior angles is always 180 degrees.

Understanding how individual angles contribute to this sum can enhance your problem-solving skills. For any triangle with angles labeled as \( A \), \( B \), and \( C \), you can always count on the equation:

• \( A + B + C = 180° \)

This relationship is crucial to solving other properties or deriving unknown variables related to triangle geometry.
Applying this knowledge, consider any variation of triangles. For example, if one angle is larger or smaller, the other two adjust accordingly so that they still sum up to 180 degrees. Recognizing this will help you spot errors and validate your geometric solutions.

Equal Angles

Two angles being equal means that they hold exactly the same measure in degrees or radians. In many problems, this condition simplifies the solution significantly because it provides a direct relationship between involved angles.

For example, if in a triangle, angles \( A \) and \( B \) are equal, then mathematically

• \( A = B \)

Furthermore, this equality allows you to represent each angle using the same variable, simplifying the calculations remarkably.
It can also suggest specific triangle types, like isosceles triangles where two equal angles typically accompany equal opposite sides as well. This results in various geometric and algebraic shortcuts during problem solving. Additionally, when angles are equal, it may simplify or present opportunities to create equivalent triangles during broader mathematical applications.

Angle Sum Property

The angle sum property of triangles states that the sum of all interior angles in a triangle is always 180 degrees. This rule is foundational to understanding and solving various geometric problems involving triangles.

Here's how this property works in practice: when you know two angles, you can easily find the third angle by subtracting the sum of known angles from 180 degrees.

• If given \( A \) and \( B \), then \( C = 180° - (A + B) \).

This principle is often the entry point for solving more complex triangle equations or verifying that a set of angles indeed form a triangle.
Using this property can help not only in direct angle calculations, but also in further understanding types of triangles and their specific characteristics. It's a dependable and universal rule that greatly simplifies geometric analysis and proof development within Euclidean geometry.