Question

if two parallel lines are cut by a transversal and one pair of corresponding angles measures 65 degrees, what is the measure of the alternate interior angle on the opposite side of the transversal?
a. 45 degrees
b. 115 degrees
c. 65 degrees
d. 90 degrees

when a transversal cuts two lines and the alternate exterior angles are congruent, what can you conclude about the lines?
a. the lines are parallel.
b. the lines are perpendicular.
c. the lines are skew.
d. the lines are neither parallel nor perpendicular.

what can be said about the opposite angles formed when two lines intersect?
a. they are congruent.
b. they are complementary.
c. they form a linear pair.
d. they are supplementary.

according to the consecutive interior (same - side interior) angles conjecture, if parallel lines m and n are cut by a transversal t, and ∠3 = 110°, what is the measure of consecutive interior angle ∠4?
a. 80°
b. 70°
c. 60°

Explanation:

Step1: Recall angle - relationships in parallel lines

When two parallel lines are cut by a transversal, corresponding angles are congruent, and alternate - interior angles are congruent. If a pair of corresponding angles measures 65 degrees, then the alternate - interior angle on the opposite side of the transversal (which is congruent to the corresponding angle) also measures 65 degrees.

Step2: Recall the condition for parallel lines

If alternate exterior angles are congruent when a transversal cuts two lines, then the two lines are parallel. This is a converse of the alternate - exterior angles theorem.

Step3: Recall vertical - angle properties

When two lines intersect, the opposite angles (vertical angles) are congruent.

Step4: Recall consecutive - interior angles property

Consecutive interior (same - side interior) angles are supplementary when two parallel lines are cut by a transversal. If one consecutive interior angle \(\angle3 = 110^{\circ}\), and let the other consecutive interior angle be \(\angle4\), then \(\angle3+\angle4 = 180^{\circ}\). So, \(\angle4=180^{\circ}-\angle3=180 - 110=70^{\circ}\).

Answer:

1. C. 65 degrees
2. A. The lines are parallel.
3. A. They are congruent.
4. B. 70°