Question

review 3.3 - 3.4 proving lines are parallel
per:
geometry
unit 3: parallel & perpendicular lines

1. given the diagram to the right and information below, determine which lines, if any, are parallel. state the converse that justifies your answer.

for questions 2 - 4, find ( x ) so that ( q parallel r ). state the converse used.
2.
3.
4.

1. ( x = ) _ ; converse: _
2. ( x = ) _ ; converse: _
3. ( x = ) _ ; converse: _

Explanation:

Step1: Solve for \(x\) in question 2

The angles \((15x + 3)^{\circ}\) and \(108^{\circ}\) are same - side exterior angles. By the converse of the same - side exterior angles theorem, if two lines are cut by a transversal and the same - side exterior angles are supplementary, then the lines are parallel.
So, \((15x+3)+108 = 180\)
\(15x+111 = 180\)
\(15x=180 - 111\)
\(15x = 69\)
\(x=\frac{69}{15}=4.6\)

Step2: Solve for \(x\) in question 3

The angles \((7x - 8)^{\circ}\) and \((11x-28)^{\circ}\) are alternate interior angles. By the converse of the alternate interior angles theorem, if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
So, \(7x-8=11x - 28\)
\(28 - 8=11x-7x\)
\(20 = 4x\)
\(x = 5\)

Step3: Solve for \(x\) in question 4

The angles \((2x)^{\circ}\) and \((5x + 21)^{\circ}\) are corresponding angles. By the converse of the corresponding angles theorem, if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.
So, \(2x+5x + 21=90\) (since the angle adjacent to the right - angle forms a right - angle with the sum of the two angles)
\(7x+21 = 90\)
\(7x=90 - 21\)
\(7x = 69\)
\(x=\frac{69}{7}\approx9.86\)

Answer:

1. \(x = 4.6\); Converse: Same - side exterior angles converse
2. \(x = 5\); Converse: Alternate interior angles converse
3. \(x=\frac{69}{7}\approx9.86\); Converse: Corresponding angles converse