Question

1. if two lines are parallel to the same line, then they are _____________ to each other.

○a perpendicular
○b congruent
○c parallel
○d bisecting

1. find the value of the variable if line m and line / are parallel lines, ( mangle1 = 2x + 44 ) and ( mangle5=5x + 38 ). the diagram is not to scale.

○a 2
○b 3
○c 4
○d 1

Explanation:

1.

This is a basic property in geometry. If line \(a\parallel m\) and line \(b\parallel m\), then by the transitive property of parallel lines (a fundamental concept in geometry), \(a\parallel b\). Perpendicular means forming a \(90^{\circ}\) angle, congruent is about having the same size and shape (not relevant for the relationship of parallelism here), and bisecting means dividing into two equal parts.

Step1: Use the property of parallel lines

When two parallel lines \(l\) and \(m\) are cut by a transversal, \(\angle1\) and \(\angle5\) are congruent (alternate - interior angles theorem). So, \(m\angle1=m\angle5\).

Step2: Set up the equation

Since \(m\angle1 = 2x + 44\) and \(m\angle5=5x + 38\), we have the equation \(2x + 44=5x+38\).

Step3: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(44 = 3x+38\).
Subtract \(38\) from both sides: \(3x=44 - 38\), so \(3x = 6\).
Divide both sides by \(3\): \(x=\frac{6}{3}=2\).

Answer:

C. Parallel

2.