. How can you prove that a constructed line is parallel to a given line? Assume that the transversal line is not perpendicular to the other lines. Show that same side exterior angles are congruent. Show that same side interior angles are congruent. Show that the transversal intersects both lines. Show that corresponding angles are congruent.
To prove that a constructed line is parallel to a given line, you can use the following steps:
1. Show that the same side exterior angles are congruent:
- Start by drawing a transversal line that intersects both the given line and the constructed line.
- Identify two exterior angles that are on the same side of the transversal line.
- Prove that these exterior angles are congruent by showing that the alternate interior angles formed by the given line and the transversal are congruent (using the property of parallel lines). Since these alternate interior angles are congruent, the same side exterior angles must also be congruent.
2. Show that the same side interior angles are congruent:
- Identify two interior angles that are on the same side of the transversal line.
- Prove that these interior angles are congruent by showing that the corresponding angles formed by the given line and the transversal are congruent (using the property of parallel lines). Since these corresponding angles are congruent, the same side interior angles must also be congruent.
3. Show that the transversal intersects both lines:
- Confirm that the constructed line and the given line both intersect the transversal line at different points. This can be observed from the diagram or explicitly stated.
4. Show that corresponding angles are congruent:
- Identify two corresponding angles formed by the given line and the transversal.
- Prove that these corresponding angles are congruent by using the property of parallel lines.
By following these steps and using the properties of parallel lines and transversals, you can prove that a constructed line is parallel to a given line.