Question

1. explain how you can use corresponding angles to prove that two lines are parallel. 6. if two parallel lines are cut by a transversal, what is the relationship between the measures of alternate interior angles? 7. a) are alternate interior angles located on the same side or opposite sides of the transversal? b) are they located on the interior or exterior of the parallel lines? a) b) 8. two parallel lines are intersected by a transversal. one of the alternate interior angles is represented by the expression (4x + 15) and the other by (6x - 25). find the value of x.

Explanation:

Step1: Recall corresponding - angles postulate

If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. That is, when we have a transversal intersecting two lines, if the angles in the same relative position (corresponding angles) have equal measures, the two lines must be parallel.

Step2: Recall alternate - interior angles theorem

When two parallel lines are cut by a transversal, the alternate - interior angles are congruent. Alternate - interior angles are pairs of angles that lie between the two parallel lines and on opposite sides of the transversal.

Step3: Answer question 7

a) Alternate - interior angles are located on opposite sides of the transversal.
b) They are located on the interior of the parallel lines.

Step4: Solve for x in question 8

Since alternate - interior angles are congruent when two parallel lines are cut by a transversal, we set the two expressions for the alternate - interior angles equal to each other:
\[4x + 15=6x−25\]
Subtract \(4x\) from both sides:
\[15 = 6x-4x - 25\]
\[15=2x - 25\]
Add 25 to both sides:
\[15 + 25=2x\]
\[40 = 2x\]
Divide both sides by 2:
\[x = 20\]

Answer:

1. If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel.
2. When two parallel lines are cut by a transversal, the alternate - interior angles are congruent.
3. a) Opposite sides of the transversal

b) Interior of the parallel lines

1. \(x = 20\)