Question

Two horizontal, parallel lines, line c and line d, where line c is above line d. These lines are intersected by two diagonal parallel lines, line a and line b. Line a is to the left of line b. The angles created by each intersection are numbered. From top left, going clockwise, the angles where line a intersects line c are labeled eleven, ten, nine, and twelve. The angles where line b intersects line c are labeled seven, six, five, and eight. The angles where line a intersects line d are labeled fifteen, fourteen, thirteen and sixteen. The angles where line b intersects line d are labeled three, two, one and four. Given ∠4≅∠14, which lines, if any, must be parallel based on the given information?

Justify your conclusion.



Responses

A. a is parallel to b, converse of the same-side interior angles theorem


B. a is parallel to b, converse of the alternate interior angles theorem


C. a is parallel to b, converse of the corresponding angles theorem


D. not enough information to make a conclusion

Answer 1

Since ∠4≅∠14 and they are corresponding angles, it justifies by the converse of the corresponding angles theorem that line a is parallel to line b.

Therefore, the correct answer is: option C). a is parallel to b, converse of the corresponding angles theorem.

Step-by-step explanation:

The question asks which lines must be parallel based on the given information that angle 4 is congruent to angle 14.

According to the properties of parallel lines intersected by a transversal, when corresponding angles are congruent, the lines being intersected by the transversal are parallel.

Properties of Parallel Lines

• They are always straight lines with an equal distance between each other.
• They are coplanar lines.
• They never intersect, no matter how far you try to extend them in any given direction.

And, Corresponding angles are the angles which are formed in matching corners or corresponding corners with the transversal when two parallel lines are intersected by any other line

In this case, Angle 4 and angle 14 are corresponding angles where line b intersects line d and line a intersects line d, respectively.

Since ∠4≅∠14, it is justified by the converse of the corresponding angles theorem that line a is parallel to line b.