Math Problem Statement

The measure of the complement of an angle excites it by 23' . Find the angle

Solution

Let's denote the measure of the angle as xx. The complement of an angle is what remains when the angle is subtracted from 9090^\circ.

The problem states that the complement of the angle exceeds the angle by 2323' (minutes). So we can write this relationship as:

(90x)=x+23(90^\circ - x) = x + 23'

Step-by-step solution:

  1. Start by setting up the equation based on the given relationship: 90x=x+2390^\circ - x = x + 23'

  2. Add xx to both sides of the equation to get rid of the negative xx on the left-hand side: 90=2x+2390^\circ = 2x + 23'

  3. Subtract 2323' from both sides: 9023=2x90^\circ - 23' = 2x

    Since 2323' is in minutes, let's convert 9090^\circ into degrees and minutes. 90=896090^\circ = 89^\circ 60', so: 896023=893789^\circ 60' - 23' = 89^\circ 37'

  4. Now we have: 8937=2x89^\circ 37' = 2x

  5. Divide both sides by 2 to solve for xx: x=89372x = \frac{89^\circ 37'}{2}

    Perform the division: x=4448.5x = 44^\circ 48.5'

So, the measure of the angle is approximately 4448.544^\circ 48.5'.

Would you like further details or explanations?

Related Questions:

  1. What is the difference between complementary and supplementary angles?
  2. How would the equation change if the complement exceeds the angle by a different amount?
  3. What happens when the angle and its complement are equal?
  4. How can angles be measured in radians instead of degrees?
  5. How do we convert minutes and seconds of an angle into decimal degrees?

Tip: Always ensure that the units (degrees and minutes) are consistent throughout your calculations.

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Math Problem Analysis

Mathematical Concepts

Complementary Angles
Angle Measurement
Algebra

Formulas

Complement of an angle = 90° - angle
Basic Algebraic Equations

Theorems

Complementary Angles Theorem

Suitable Grade Level

Grades 7-10