Math Problem Statement

Two or more angles are complementary if their sum is 90°. Three angles are complementary. One angle is three times the value of the smallest angle. The largest angle is five times the value of the smallest angle. Find the measures of the three angles.

Solution

To solve the problem:

Definitions and Setup

Let:

  • xx = the smallest angle.
  • 3x3x = the angle that is three times the smallest.
  • 5x5x = the largest angle, which is five times the smallest.

Complementary Rule

The sum of the three angles is 9090^\circ. Hence, we set up the equation: x+3x+5x=90x + 3x + 5x = 90

Simplification

9x=909x = 90

Solve for xx

x=909=10x = \frac{90}{9} = 10

Values of the Angles

  • Smallest angle: x=10x = 10^\circ
  • Second angle: 3x=303x = 30^\circ
  • Largest angle: 5x=505x = 50^\circ

Final Answer

The three angles are 1010^\circ, 3030^\circ, and 5050^\circ.

Let me know if you want further details or additional explanations. Here are 5 related questions to expand your understanding:

  1. What is the definition of complementary angles?
  2. How do you solve similar problems with more unknown variables?
  3. Can complementary angles involve more than three angles?
  4. What happens if one of the angles is expressed as a fraction of another?
  5. How does this change if the total sum were 180180^\circ (supplementary angles)?

Tip: Always check the sum of your angles to ensure the calculations align with the given conditions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Complementary Angles

Formulas

Sum of complementary angles: x + 3x + 5x = 90

Theorems

Definition of complementary angles

Suitable Grade Level

Grades 6-8