Math Problem Statement

∠1 and angle, 2∠2 are complementary angles. If m, angle, 1, equals, left parenthesis, x, minus, 22, right parenthesis, degreesm∠1=(x−22) ∘ and m, angle, 2, equals, left parenthesis, x, minus, 16, right parenthesis, degreesm∠2=(x−16) ∘ , then find the measure of angle, 1∠1

Solution

Complementary angles are two angles whose measures add up to 9090^\circ. Given that 1\angle 1 and 2\angle 2 are complementary, we can write the equation:

m1+m2=90m\angle 1 + m\angle 2 = 90^\circ

Substituting the given expressions for m1m\angle 1 and m2m\angle 2:

(x22)+(x16)=90(x - 22) + (x - 16) = 90

Now, let's solve for xx:

x22+x16=90x - 22 + x - 16 = 90

Combine like terms:

2x38=902x - 38 = 90

Add 38 to both sides:

2x=1282x = 128

Divide by 2:

x=64x = 64

Now that we have the value of xx, we can find the measure of 1\angle 1 by substituting xx back into the expression for m1m\angle 1:

m1=x22=6422=42m\angle 1 = x - 22 = 64 - 22 = 42^\circ

Answer:

The measure of 1\angle 1 is 4242^\circ.

Would you like more details or have any questions? Here are some related questions to explore:

  1. How do you find the measure of complementary angles if one angle is given?
  2. What are supplementary angles, and how are they different from complementary angles?
  3. If the measure of 1\angle 1 was given in radians, how would the calculation change?
  4. How do you find the value of xx if both angles are equal?
  5. How would you solve this problem if you had three angles that add up to 180180^\circ?

Tip: When solving problems involving complementary or supplementary angles, always remember the key relationships: 9090^\circ for complementary and 180180^\circ for supplementary.

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Math Problem Analysis

Mathematical Concepts

Complementary Angles
Linear Equations

Formulas

m∠1 + m∠2 = 90°
Solving linear equations

Theorems

Complementary Angle Theorem

Suitable Grade Level

Grades 7-9