Examkin

Question

Difficulty: MediumLines and Angles

A laser cutting machine is programmed to make two straight intersecting cuts on a sheet of acrylic. The measures of a pair of vertical angles created by the intersection are (9x−18)∘(9x - 18)^\circ and (5x+46)∘(5x + 46)^\circ. What is the measure of the acute angle formed by these cuts, in degrees?

Answer: 54 degrees

Answer

The measure of the acute angle is 54 degrees.
Setting the expressions for the vertical angles equal to each other gives 9x−18=5x+469x - 18 = 5x + 46. Solving for xx yields x=16x = 16. Substituting 1616 back into the expression (9x−18)∘(9x - 18)^\circ results in a vertical angle of 126∘126^\circ. Since the intersecting lines form supplementary adjacent angles, the adjacent angle is 180∘−126∘=54∘180^\circ - 126^\circ = 54^\circ. This represents the acute angle.

Step-by-Step Solution

1
Set up an equation by equating the two algebraic expressions.
9x−18=5x+469x - 18 = 5x + 46
Vertical angles formed by intersecting lines are always equal in measure.
2
Solve the equation for the variable xx.
4x=64⇒x=164x = 64 \Rightarrow x = 16
Isolating xx provides the numerical value needed to evaluate the angle expressions.
3
Substitute x=16x = 16 back into either original expression.
9(16)−18=144−18=126∘9(16) - 18 = 144 - 18 = 126^\circ
Evaluating the expression gives the actual degree measure of the vertical angles.
4
Calculate the measure of the adjacent acute angle.
180∘−126∘=54∘180^\circ - 126^\circ = 54^\circ
Adjacent angles on a straight line are supplementary (add up to 180∘180^\circ). Because 54∘<90∘54^\circ < 90^\circ, this is the required acute angle.

Key Concept

Vertical angles are equal, and adjacent angles formed by intersecting lines are supplementary.
Rate this question

Topics

Lines and Angles

More questions from this topic