Question

mathematical practices
do you understand?

1. vocabulary why is the word coplanar included in the definition for parallel lines?
2. vocabulary how does the phrase alternate interior angles describe the positions of the two angles?
3. error analysis in the figure at the right, lines and planes that appear to be parallel are parallel. carly says (overline{ab}paralleloverline{hg}). juan says (overline{ab}) and (overline{hg}) are skew. who is correct? explain.

Explanation:

Step1: Recall coplanar definition

Coplanar means lying in the same plane. For parallel lines, they are coplanar because they are in the same flat - two - dimensional surface.

Step2: Analyze alternate interior angles

Alternate interior angles are non - adjacent angles that lie between two lines and on opposite sides of a transversal. In a three - dimensional figure like a cube, when considering parallel lines and a transversal, these angles have specific relative positions.

Step3: Evaluate statements about lines and planes

In a cube, $\overline{AB}$ and $\overline{HG}$ are parallel. They lie in parallel planes (the front and back faces of the cube). They are not skew. Skew lines are non - parallel and non - coplanar. Since $\overline{AB}$ and $\overline{HG}$ are parallel and coplanar in a sense related to the parallel planes they are part of, Juan is correct.

Answer:

1. Parallel lines are coplanar because they lie in the same plane.
2. Alternate interior angles are non - adjacent angles that lie between two lines and on opposite sides of a transversal.
3. Juan is correct. $\overline{AB}$ and $\overline{HG}$ are parallel as they lie in parallel planes and are not skew.