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Chapter 3: Problem 9
In triangle \(A B C,\) the measure of angle \(B\) is three times that of angle \(A.\) The measure of angle \(C\) is \(20^{\circ}\) more than that of angle \(A .\) Findthe angle measures.
Short Answer
Expert verified
Angle A is 32°; Angle B is 96°; Angle C is 52°.
Step by step solution
01
Represent the Angles with Variables
Let the measure of angle A be denoted as \(x\). Therefore, the measure of angle B, which is three times angle A, can be represented as \(3x\). The measure of angle C, which is 20 degrees more than angle A, can be represented as \(x + 20^{\circ}\).
02
Set Up the Equation
The sum of the angles in any triangle is always \(180^{\circ}\). Therefore, set up the equation combining all the angle measures: \[x + 3x + (x + 20^{\circ}) = 180^{\circ}\] Simplify the equation to solve for \(x\).
03
Simplify the Equation
Combine the like terms in the equation: \[x + 3x + x + 20^{\circ} = 180^{\circ}\] This simplifies to: \[5x + 20^{\circ} = 180^{\circ}\]
04
Solve for x
Subtract \(20^{\circ}\) from both sides of the equation: \[5x = 160^{\circ}\] Divide both sides by 5: \[x = 32^{\circ}\]
05
Determine All Angles
Now that we know \(x = 32^{\circ}\), we can find the measures of the other angles. Angle A is \(32^{\circ}\), angle B is \3x = 3(32^{\circ}) = 96^{\circ}\, and angle C is \x + 20^{\circ} = 32^{\circ} + 20^{\circ} = 52^{\circ}\.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
angles in a triangle
Understanding the angles in a triangle is crucial for solving many geometry problems. A triangle has three angles. The measures of these angles always add up to 180 degrees. This is a fundamental property of all triangles. When solving problems involving triangle angles, it's essential to remember this rule because it allows you to create equations that help find unknown angles. For example, if you know two angles, you can easily find the third by subtracting the sum of the known angles from 180 degrees.
solving equations
Solving equations is a key skill in algebra and geometry. An equation sets up a relationship between unknown values (variables) and known values. To solve an equation, follow these steps:
- Combine like terms to simplify the equation.
- Use inverse operations to isolate the variable.
- Perform the same operations on both sides of the equation to maintain equality.
In our example, the equation is set up using the angle sum property of a triangle. After simplification, we isolate the variable by first subtracting and then dividing to find the value of the unknown angle.
variable representation
Variable representation helps to model quantities and their relationships. In our problem, we let the measure of angle A be denoted as x. This simplifies writing and solving our equations.
- The measure of angle B, which is three times the measure of angle A, is represented as 3x.
- The measure of angle C, which is 20 degrees more than angle A, is represented as x + 20 degrees.
By representing angles as variables, you create a system that can be manipulated mathematically to solve for unknown values.
angle sum property
The angle sum property is an essential rule in triangle geometry. It states that the sum of the measures of the angles in a triangle is always 180 degrees. This property is foundational and allows for the formulation of equations to solve for unknown angles.
In our exercise, we used this property by setting up the following equation involving the variable representations: x + 3x + (x + 20) = 180. By simplifying and solving this equation, we found the measures of all three angles in the triangle.
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