Two Angles Are Supplementary, With One Of The Angles Being 30° Less Than Five Times The Other. Find The Measures Of The Two Angles.A. 35° And 145° B. 28° And 152° C. 40° And 140° D. 68° And 112°

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Introduction

In geometry, supplementary angles are two angles whose measures add up to 180°. In this article, we will explore a problem involving supplementary angles, where one angle is 30° less than five times the other. We will use algebraic methods to solve for the measures of the two angles.

Understanding the Problem

Let's denote the measure of the smaller angle as x. Since the larger angle is 30° less than five times the smaller angle, its measure can be expressed as 5x - 30. We know that the sum of the measures of supplementary angles is 180°. Therefore, we can set up the following equation:

x + (5x - 30) = 180

Solving the Equation

To solve for x, we need to combine like terms and isolate the variable. Let's start by combining the x terms:

x + 5x = 6x

Now, let's rewrite the equation with the combined x terms:

6x - 30 = 180

Next, let's add 30 to both sides of the equation to get rid of the negative term:

6x = 210

Now, let's divide both sides of the equation by 6 to solve for x:

x = 35

Finding the Measures of the Angles

Now that we have found the measure of the smaller angle (x = 35), we can find the measure of the larger angle by substituting x into the expression 5x - 30:

5(35) - 30 = 175 - 30 = 145

Therefore, the measures of the two supplementary angles are 35° and 145°.

Conclusion

In this article, we used algebraic methods to solve a problem involving supplementary angles. We set up an equation based on the given information, solved for the variable, and found the measures of the two angles. The correct answer is A. 35° and 145°.

Additional Examples and Practice

If you want to practice solving supplementary angle problems, try the following examples:

  • Two angles are supplementary, with one angle being 20° less than three times the other. Find the measures of the two angles.
  • Two angles are supplementary, with one angle being 15° more than twice the other. Find the measures of the two angles.

Answer Key

A. 35° and 145° B. 28° and 152° C. 40° and 140° D. 68° and 112°

Mathematical Concepts

  • Supplementary angles: two angles whose measures add up to 180°
  • Algebraic methods: using equations and variables to solve problems
  • Solving equations: combining like terms, isolating variables, and solving for unknowns

Real-World Applications

  • Architecture: designing buildings and structures that require precise angle measurements
  • Engineering: creating machines and mechanisms that rely on accurate angle calculations
  • Art and design: creating visual compositions that involve geometric shapes and angles

Tips and Tricks

  • When solving supplementary angle problems, make sure to set up the equation correctly and combine like terms.
  • Use algebraic methods to solve for the variable, and then find the measures of the two angles.
  • Practice solving supplementary angle problems to build your skills and confidence.
    Supplementary Angles Q&A: Frequently Asked Questions =====================================================

Introduction

In our previous article, we explored the concept of supplementary angles and solved a problem involving two angles whose measures add up to 180°. In this article, we will answer some frequently asked questions about supplementary angles.

Q: What are supplementary angles?

A: Supplementary angles are two angles whose measures add up to 180°. They can be adjacent or non-adjacent, and they can be acute, right, or obtuse.

Q: How do I find the measures of supplementary angles?

A: To find the measures of supplementary angles, you can use algebraic methods. Set up an equation based on the given information, solve for the variable, and then find the measures of the two angles.

Q: What is the difference between supplementary and complementary angles?

A: Supplementary angles are two angles whose measures add up to 180°, while complementary angles are two angles whose measures add up to 90°.

Q: Can supplementary angles be negative?

A: No, supplementary angles cannot be negative. The sum of the measures of two angles must be 180°, and negative angles do not exist in geometry.

Q: Can supplementary angles be zero?

A: No, supplementary angles cannot be zero. The sum of the measures of two angles must be 180°, and zero is not a valid angle measure.

Q: How do I know if two angles are supplementary?

A: To determine if two angles are supplementary, add their measures together. If the sum is 180°, then the angles are supplementary.

Q: Can supplementary angles be equal?

A: Yes, supplementary angles can be equal. For example, two angles that measure 90° each are supplementary.

Q: What are some real-world applications of supplementary angles?

A: Supplementary angles have many real-world applications, including:

  • Architecture: designing buildings and structures that require precise angle measurements
  • Engineering: creating machines and mechanisms that rely on accurate angle calculations
  • Art and design: creating visual compositions that involve geometric shapes and angles

Q: How can I practice solving supplementary angle problems?

A: You can practice solving supplementary angle problems by:

  • Using online resources and worksheets
  • Creating your own problems and solutions
  • Working with a partner or tutor to practice and review

Conclusion

In this article, we answered some frequently asked questions about supplementary angles. We covered topics such as the definition of supplementary angles, how to find their measures, and real-world applications. We hope this article has been helpful in clarifying any questions you may have had about supplementary angles.

Additional Resources

  • Online resources and worksheets for supplementary angle problems
  • Video tutorials and explanations of supplementary angle concepts
  • Practice problems and solutions for supplementary angle exercises

Supplementary Angle Formulas

  • The sum of the measures of two supplementary angles is 180°: m∠1 + m∠2 = 180°
  • The difference between the measures of two supplementary angles is 180°: m∠1 - m∠2 = 180°

Supplementary Angle Examples

  • Two angles are supplementary, with one angle being 20° less than three times the other. Find the measures of the two angles.
  • Two angles are supplementary, with one angle being 15° more than twice the other. Find the measures of the two angles.

Answer Key

A. 35° and 145° B. 28° and 152° C. 40° and 140° D. 68° and 112°