What Is The Difference In The Measures Of Two Supplementary Angles Is 38?

The two supplementary angles are 109° and 71°.

Understanding Supplementary Angles

Supplementary angles are a fundamental concept in geometry that describes a special relationship between two angles. By definition, two angles are considered supplementary if the sum of their individual measures is exactly 180°. This concept is often visualized as two angles that, when placed adjacent to each other, form a straight line. For example, if we have two angles, let's call them Angle A and Angle B, they are supplementary if their measures satisfy the equation:

Angle A+Angle B=180∘Angle A plus Angle B equals 180 raised to the composed with power

AngleA+AngleB=180∘

It's important to distinguish supplementary angles from complementary angles, which are two angles that add up to 90°.

The Problem: Finding the Angles

The problem states that we have two supplementary angles, and the difference in their measures is 38°. To solve this, we can use a system of linear equations, which is a powerful algebraic method for solving problems with multiple unknown variables.

Setting Up the Equations

First, let's represent the two unknown angle measures with variables. We can use xx

𝑥

and yy

𝑦

to represent the measure of each angle. Based on the problem's information, we can create two equations:

1. Supplementary Angle Equation: Since the angles are supplementary, their sum is 180°.x+y=180x plus y equals 180

   𝑥+𝑦=180

2. Difference Equation: The difference between the two angles is 38°. To ensure a positive difference, we can assume xx

   𝑥

   is the larger angle.x−y=38x minus y equals 38

   𝑥−𝑦=38

Solving the System of Equations

Now, we have a system of two equations with two variables. A straightforward way to solve this is using the elimination method. By adding the two equations together, we can eliminate one of the variables.

1. Add the two equations: We'll add the left side of both equations and the right side of both equations.(x+y)+(x−y)=180+38open paren x plus y close paren plus open paren x minus y close paren equals 180 plus 38

   (𝑥+𝑦)+(𝑥−𝑦)=180+38

2. Simplify the equation: The +ypositive y

   +𝑦

   and −ynegative y

   −𝑦

   terms cancel each other out, leaving us with a single variable, xx

   𝑥

   .2x=2182 x equals 218

   2𝑥=218

3. Solve for x: To find the value of xx

   𝑥

   , we divide both sides by 2.x=2182x equals 218 over 2 end-fraction

   𝑥=2182

   x=109x equals 109

   𝑥=109

We have now found the measure of the first angle, which is 109°.

Finding the Second Angle

With the value of xx

𝑥

known, we can substitute it back into either of our original equations to find the value of yy

𝑦

. Using the first equation is often the simplest approach:

1. **Substitute x into the first equation:**109+y=180109 plus y equals 180

   109+𝑦=180

2. Solve for y: To isolate yy

   𝑦

   , we subtract 109 from both sides of the equation.y=180−109y equals 180 minus 109

   𝑦=180−109

   y=71y equals 71

   𝑦=71

The measure of the second angle is 71°.

Verification and Final Answer

Finally, it's good practice to verify our answers by checking if they satisfy the initial conditions of the problem.

1. **Are they supplementary?**109∘+71∘=180∘109 raised to the composed with power plus 71 raised to the composed with power equals 180 raised to the composed with power

   109∘+71∘=180∘

   Yes, they are supplementary.

2. **Is their difference 38°?**109∘−71∘=38∘109 raised to the composed with power minus 71 raised to the composed with power equals 38 raised to the composed with power

   109∘−71∘=38∘

   Yes, their difference is 38°.

Both conditions are met, confirming that our calculations are correct. The two angles are 109° and 71°.

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