Finding Angle 2 The Measure Of Angle 1 Is (10x+8) And Angle 3 Is (12x-10)
In the realm of geometry, understanding the relationships between angles is crucial for solving various problems. This article delves into a specific scenario involving angles 1, 2, and 3, where the measures of angles 1 and 3 are expressed in terms of a variable 'x'. Our primary goal is to determine the measure of angle 2 in degrees. To achieve this, we will leverage fundamental geometric principles, particularly the properties of angles formed by intersecting lines and the concept of supplementary angles. Let's embark on this geometric exploration and unravel the value of angle 2.
Identifying Angle Relationships
Before we dive into the calculations, it's essential to establish the relationship between angles 1, 2, and 3. Without a visual diagram, we'll assume a common geometric configuration where angles 1 and 3 are vertical angles and angle 2 is supplementary to both angles 1 and 3. Vertical angles are formed when two lines intersect, and they are always equal in measure. Supplementary angles, on the other hand, are two angles that add up to 180 degrees. This understanding forms the bedrock of our solution.
Considering this relationship, the measure of angle 1 (10x + 8)° and the measure of angle 3 (12x - 10)° provide the necessary information to set up an equation and solve for 'x'. Since vertical angles are equal, we can equate the expressions representing their measures: (10x + 8)° = (12x - 10)°. This equation allows us to determine the value of 'x', which in turn will help us calculate the measures of angles 1 and 3. Once we know the measure of either angle 1 or angle 3, we can easily find the measure of angle 2 using the supplementary angle relationship. Angle 2, being supplementary to both angle 1 and angle 3, will have a measure that, when added to either angle 1 or angle 3, equals 180 degrees. This is the core concept we'll use to find our final answer.
Solving for 'x'
To find the measure of angle 2, we must first determine the value of 'x'. As established earlier, angles 1 and 3 are vertical angles and therefore equal. This gives us the equation:
(10x + 8)° = (12x - 10)°
To solve for 'x', we need to isolate the variable on one side of the equation. Let's start by subtracting 10x from both sides:
8 = 2x - 10
Next, add 10 to both sides:
18 = 2x
Finally, divide both sides by 2:
x = 9
Now that we have the value of 'x', we can substitute it back into the expressions for angles 1 and 3 to find their measures. This step is crucial as it bridges the gap between the algebraic representation of the angles and their numerical values in degrees. By substituting x = 9, we transform the expressions (10x + 8)° and (12x - 10)° into concrete angle measurements, allowing us to proceed with finding the measure of angle 2. The accuracy of this substitution is paramount, as any error here will propagate through the rest of the solution. We will then use these measures and the supplementary angle property to calculate the measure of angle 2.
Calculating the Measures of Angles 1 and 3
With x = 9, we can now calculate the measures of angles 1 and 3. This step is vital because it provides the numerical values needed to determine the measure of angle 2, which is supplementary to both angles 1 and 3. We'll substitute the value of x into the expressions we have for the angles.
For angle 1:
(10x + 8)° = (10 * 9 + 8)° = (90 + 8)° = 98°
For angle 3:
(12x - 10)° = (12 * 9 - 10)° = (108 - 10)° = 98°
As expected, angles 1 and 3 have the same measure, which confirms our initial understanding that they are vertical angles. This equality serves as a check on our calculations and reinforces the geometric principle that vertical angles are congruent. Now that we know the measures of angles 1 and 3, we can move on to the final step of finding the measure of angle 2. The supplementary relationship between angle 2 and either angle 1 or angle 3 will be the key to unlocking the solution. We will use this relationship to set up an equation and solve for the measure of angle 2.
Determining the Measure of Angle 2
Since angle 2 is supplementary to angle 1 (and angle 3), the sum of their measures is 180°. We can use this information to find the measure of angle 2. Let's denote the measure of angle 2 as 'y'. Then, we have:
y + 98° = 180°
To solve for 'y', subtract 98° from both sides:
y = 180° - 98°
y = 82°
Therefore, the measure of angle 2 is 82°. This result is the culmination of our step-by-step approach, which involved understanding angle relationships, solving for an unknown variable, and applying the concept of supplementary angles. We can verify this result by adding the measure of angle 2 to the measure of either angle 1 or angle 3; the sum should be 180°. This verification step is a crucial part of problem-solving, as it ensures the accuracy of our calculations and the validity of our solution. In this case, 82° + 98° = 180°, confirming that our answer is correct.
Conclusion
In conclusion, by applying geometric principles and algebraic techniques, we have successfully determined that the measure of angle 2 is 82°. This problem highlights the importance of understanding angle relationships, such as vertical and supplementary angles, in solving geometric problems. The step-by-step approach, including solving for 'x' and substituting its value, demonstrates a systematic method for tackling such problems. This methodology is not only applicable to this specific problem but also serves as a valuable framework for solving a wide range of geometric challenges. The ability to identify angle relationships, set up equations, and solve for unknowns is a fundamental skill in geometry and a testament to the power of mathematical reasoning. By mastering these concepts, students can confidently approach and solve complex geometric problems, expanding their understanding of the world around them. This exercise not only provides a solution to a specific problem but also reinforces the broader applicability of geometric principles in various fields and disciplines.
Final Answer: The measure of angle 2 is 82 degrees.
