Question

1. explain how you can use the concept of alternate interior angles to determine if two lines are parallel.
2. if two parallel lines are cut by a transversal, what is the relationship between the measures of alternate exterior angles?
3. a) are alternate exterior 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?
4. two parallel lines are intersected by a transversal. one of the alternate exterior angles is represented by the expression (5y - 30) and the other by (2y + 60). find the value of y. show your work below:

Explanation:

Step1: Recall alternate - interior angles property

If two lines are cut by a transversal, and the alternate - interior angles are congruent, then the two lines are parallel. That is, if we have two lines \(l_1\) and \(l_2\) cut by a transversal \(t\), and the pairs of alternate - interior angles \(\angle1\) and \(\angle2\) (or the other pair) are equal in measure (\(\angle1=\angle2\)), then \(l_1\parallel l_2\).

Step2: Recall alternate - exterior angles property for parallel lines

When two parallel lines are cut by a transversal, the alternate - exterior angles are congruent. That is, if \(l_1\parallel l_2\) and \(t\) is the transversal, the pairs of alternate - exterior angles have the same measure.

Step3: Answer question 11

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

Step4: Solve for \(y\) in question 12

Since the alternate - exterior angles are congruent when two parallel lines are cut by a transversal, we set the two angle expressions equal to each other:
\[5y - 30=2y+60\]
Subtract \(2y\) from both sides:
\[5y-2y - 30=2y - 2y+60\]
\[3y-30 = 60\]
Add 30 to both sides:
\[3y-30 + 30=60 + 30\]
\[3y=90\]
Divide both sides by 3:
\[y = 30\]

Answer:

1. If the alternate - interior angles formed by two lines and a transversal are congruent, the two lines are parallel.
2. The alternate - exterior angles are congruent.
3. a) Opposite sides. b) Exterior.
4. \(y = 30\)