Find the measure of each marked angle. See Example 2 complementary angles with measures 3𝓍 ― 5 and 6𝓍 ― 40 degrees

Verified step by step guidance

1. Recall that complementary angles are two angles whose measures add up to 90 degrees. So, if the two angles are \$3x - 5\( and \)6x - 40$, their sum is \(90\) degrees.
2. Set up the equation representing the sum of the complementary angles: \[(3x - 5) + (6x - 40) = 90\]
3. Combine like terms on the left side: \[3x + 6x - 5 - 40 = 90\] which simplifies to \[9x - 45 = 90\]
4. Solve for \(x\) by first adding 45 to both sides: \[9x = 90 + 45\] then divide both sides by 9: \[x = \frac{135}{9}\]
5. Once you find the value of \(x\), substitute it back into the expressions for each angle: \[3x - 5\] and \[6x - 40\] to find their measures.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. Understanding this relationship allows you to set up an equation where the sum of the given angle expressions equals 90, which is essential for solving for the variable.

Algebraic Expressions in Angle Measures

Angles can be represented using algebraic expressions involving variables. To find the actual angle measures, you need to form and solve equations based on these expressions, applying algebraic techniques such as combining like terms and isolating variables.

Solving Linear Equations

Solving for the variable in the angle expressions requires knowledge of linear equations. This involves manipulating the equation formed by the sum of the angles to find the value of the variable, which can then be substituted back to find each angle's measure.