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Chapter 3: Problem 10

In triangle \(A B C,\) the measure of angle \(B\) is twice the measure of angle \(A .\) The measure of angle \(C\) is \(80^{\circ}\) more than that of angle \(A .\) Find the angle measures.

Short Answer

Expert verified
Angle A = 25掳, Angle B = 50掳, Angle C = 105掳.

Step by step solution

01

- Understand the Given Information

In the triangle ABC, it is given that: 1) The measure of angle B is twice the measure of angle A. 2) The measure of angle C is 80掳 more than the measure of angle A.
02

- Set Up the Equations

Let the measure of angle A be represented by the variable x. Then: Angle B = 2x, Angle C = x + 80掳.
03

- Use the Triangle Angle Sum Property

The sum of all angles in a triangle is 180掳. Therefore, you have: x + 2x + (x + 80掳) = 180掳
04

- Simplify the Equation

Combine like terms: 4x + 80掳 = 180掳
05

- Solve for x

Subtract 80掳 from both sides to get: 4x = 100掳 Then, divide by 4: x = 25掳
06

- Find the Measure of Each Angle

Angle A = x = 25掳 Angle B = 2x = 2(25掳) = 50掳 Angle C = x + 80掳 = 25掳 + 80掳 = 105掳

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

triangle properties
Triangles are fundamental shapes in geometry that have three sides, three vertices, and three interior angles. These angles always interact in predictable ways. Understanding the properties of triangles is crucial for solving angle-related problems. In any triangle, the interior angles always add up to a specific value, which is 180 degrees. This rule helps in finding unknown angles when some angles are known. Additionally, triangles come in various types such as equilateral, isosceles, and scalene, all having unique properties, but they all share this angle sum property.
angle sum property
The angle sum property of a triangle is a vital theorem in geometry. It states that the sum of the interior angles of a triangle is always 180 degrees. This can be written mathematically as: \[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\text{o} \]. This property is used to solve many problems, where not all the angle measures are known initially. Knowing two angles can directly lead to finding the third one by subtracting the sum of the two known angles from 180 degrees. For example, in the given problem, the angle sum property was crucial in setting up the equation for the unknown angles.
algebraic equations
Algebraic equations are mathematical statements that show the equality between two expressions involving variables. In the context of triangles, these equations help in representing unknown angle measures in terms of variables. For instance, if Angle A is represented as \( x \), other angles can be expressed relative to \( x \) using given relationships. In this exercise, Angle B, being twice the measure of Angle A, is written as \( 2x \), and Angle C, being 80 degrees more than Angle A, is written as \( x + 80^\text{o} \). By substituting these expressions into the angle sum equation, we can solve for \( x \).
variable substitution
Variable substitution is a method used in algebra to solve equations by replacing variables with their corresponding values. In our exercise, once the variable \( x \) is determined using the equation derived from the angle sum property, we can substitute it back into the expressions for Angles B and C to find their measures. After solving \( 4x + 80^\text{o} = 180^\text{o} \) to find \( x \), the value 25 degrees is substituted back into \( 2x \) and \( x + 80^\text{o} \) to find that Angle B is 50 degrees and Angle C is 105 degrees. This method simplifies solving the problem systematically.

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