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Zorluk: OrtaLines 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?

Cevap: 54 degrees

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Vertical angles are equal, and adjacent angles formed by intersecting lines are supplementary.
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Konular

Lines and Angles

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