Question

1. 10) find the value of x that makes lines u and v parallel. 11) 12) 13) 14) 15) 16) critical thinking questions: 17) for question #16, find a value of x that makes lines u and v intersect. 18) even if the lines in question #16 were not parallel, could x = 25? why or why not?

Explanation:

Step1: Recall parallel - line angle relationships

For two lines \(u\) and \(v\) cut by a transversal, we use corresponding - angles, alternate - interior angles, or same - side interior angles relationships. If the lines are parallel, corresponding angles are equal, alternate - interior angles are equal, and same - side interior angles are supplementary.

Question 11

The angles \(16x - 6\) and \(90^{\circ}\) are corresponding angles. For lines \(u\) and \(v\) to be parallel, \(16x-6 = 90\).

Step2: Solve the equation for \(x\)

Add 6 to both sides of the equation \(16x-6 = 90\):
\(16x=90 + 6=96\).
Divide both sides by 16: \(x=\frac{96}{16}=6\).

Question 12

The angles \(12x - 4\) and \(10x+10\) are corresponding angles. For lines \(u\) and \(v\) to be parallel, \(12x-4=10x + 10\).

Step2: Solve the equation for \(x\)

Subtract \(10x\) from both sides: \(12x-10x-4=10x-10x + 10\), which simplifies to \(2x-4 = 10\).
Add 4 to both sides: \(2x=10 + 4=14\).
Divide both sides by 2: \(x = 7\).

Question 13

The angles \(105^{\circ}\) and \(x + 112\) are same - side interior angles. For lines \(u\) and \(v\) to be parallel, \(105+(x + 112)=180\).

Step2: Solve the equation for \(x\)

First, simplify the left - hand side: \(x+217 = 180\).
Subtract 217 from both sides: \(x=180 - 217=-37\).

Question 14

The angles \(19x-4\) and \(110^{\circ}\) are corresponding angles. For lines \(u\) and \(v\) to be parallel, \(19x-4 = 110\).

Step2: Solve the equation for \(x\)

Add 4 to both sides: \(19x=110 + 4=114\).
Divide both sides by 19: \(x = 6\).

Question 15

The angles \(11x-2\) and \(130^{\circ}\) are corresponding angles. For lines \(u\) and \(v\) to be parallel, \(11x-2=130\).

Step2: Solve the equation for \(x\)

Add 2 to both sides: \(11x=130 + 2=132\).
Divide both sides by 11: \(x = 12\).

Question 16

The angles \(7x + 14\) and \(8x+6\) are corresponding angles. For lines \(u\) and \(v\) to be parallel, \(7x + 14=8x+6\).

Step2: Solve the equation for \(x\)

Subtract \(7x\) from both sides: \(7x-7x + 14=8x-7x+6\), which simplifies to \(14=x + 6\).
Subtract 6 from both sides: \(x = 8\).

Question 17

For lines \(u\) and \(v\) to intersect in question 16, we want \(7x + 14
eq8x+6\).
Let's find when they are not equal. Solving \(7x + 14=8x+6\) gives \(x = 8\). So for \(x
eq8\), the lines will intersect. For example, \(x = 9\) (any value other than 8 will work).

Question 18

In question 16, if \(x = 25\), the angle on line \(u\) is \(7x+14=7\times25+14=175 + 14=189\) and the angle on line \(v\) is \(8x+6=8\times25+6=200 + 6=206\). Since angles formed by two lines and a transversal cannot be greater than \(180^{\circ}\) in the context of Euclidean geometry, \(x\) cannot be 25.

Answer:

1. \(x = 6\)
2. \(x = 7\)
3. \(x=-37\)
4. \(x = 6\)
5. \(x = 12\)
6. \(x = 8\)
7. \(x

eq8\) (e.g., \(x = 9\))

1. No, because angles formed by two lines and a transversal cannot be greater than \(180^{\circ}\) in Euclidean geometry.