Lines k and l are parallel. Lines m and n intersect lines k and l to form 2 triangles. The top triangle has angles 1, 2, 3. The bottom triangle has angles 4, 5, 6.

Question

Given k Is parallel to l, what can you conclude about these triangles? Check all that apply.
Vertical angles prove Angle1 Is congruent to Angle4.
Alternate exterior angles prove Angle5 Is congruent to Angle3.
Angle5 and Angle3 are alternate interior angles.
The ratios of the corresponding angles are equivalent.
The triangles are similar because all corresponding angles are congruent.

Answer 1

Given that lines \( k \) and \( l \) are parallel and lines \( m \) and \( n \) intersect these lines, we can analyze the angles formed in the two triangles.

1. Vertical angles prove Angle 1 is congruent to Angle 4.

   • This is true. Vertical angles are equal, so if Angle 1 and Angle 4 are vertical angles, they are congruent.

2. Alternate exterior angles prove Angle 5 is congruent to Angle 3.

   • This statement is not directly valid unless it can be confirmed that Angle 5 and Angle 3 are alternate exterior angles. However, if lines \( m \) and \( n \) are transversal lines intersecting lines \( k \) and \( l \), and based on the position described in the problem, this could potentially be valid, but further information about their arrangement is needed.

3. Angle 5 and Angle 3 are alternate interior angles.

   • This is true if lines \( m \) and \( n \) act as transversals intersecting the parallel lines. If so, then Angle 5 and Angle 3 are indeed alternate interior angles and therefore are congruent.

4. The ratios of the corresponding angles are equivalent.

   • This is true. Since the angles of the triangles involve corresponding angles created by the transversals intersecting the parallel lines, and if Angle 1 is congruent to Angle 4, Angle 2 to Angle 5, and Angle 3 to Angle 6, then the ratios of the corresponding angles will be equal.

5. The triangles are similar because all corresponding angles are congruent.

   • This is true. If all corresponding angles of the triangles are congruent (via the properties of angles formed by transversals cutting across parallel lines), then the triangles are similar by the Angle-Angle (AA) similarity criterion.

In conclusion, the valid statements based on the properties of parallel lines and transversal angles are:

• Vertical angles prove Angle 1 is congruent to Angle 4.
• Angle 5 and Angle 3 are alternate interior angles.
• The ratios of the corresponding angles are equivalent.
• The triangles are similar because all corresponding angles are congruent.

The statement regarding alternate exterior angles needs verification based on the specific arrangement of the angles.