Question

1. in the figure, the lines are parallel. what is the relationship given?

o supplementary
o vertical
acute angle
obtuse angle:

Explanation:

Step1: Use the property of parallel lines and transversal

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Here, \(42x + 4\) and \(44x-2\) are consecutive interior angles. So, \((42x + 4)+(44x-2)=180\).

Step2: Simplify the equation

Combine like - terms: \(42x+44x+4 - 2=180\), which gives \(86x+2 = 180\).
Subtract 2 from both sides: \(86x=180 - 2=178\).
Divide both sides by 86: \(x=\frac{178}{86}=\frac{89}{43}\approx2.07\).

Step3: Find the measures of the angles

For the angle \(42x + 4\): substitute \(x = \frac{89}{43}\), \(42\times\frac{89}{43}+4=\frac{3738}{43}+4=\frac{3738 + 172}{43}=\frac{3910}{43}\approx90.93^{\circ}\) (obtuse angle).
For the angle \(44x-2\): substitute \(x=\frac{89}{43}\), \(44\times\frac{89}{43}-2=\frac{3916}{43}-2=\frac{3916-86}{43}=\frac{3830}{43}\approx89.07^{\circ}\) (acute angle).

Answer:

Acute Angle: \(\frac{3830}{43}\approx89.07^{\circ}\), Obtuse Angle: \(\frac{3910}{43}\approx90.93^{\circ}\)