Question

given: \\( \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } \\) and \\( \overleftrightarrow { e f } \\) intersects \\( \overleftrightarrow { a b } \\) at point \\( h \\) and intersects \\( \overleftrightarrow { c d } \\) at point \\( k \\).
prove: \\( \angle h k d \cong \angle e h b \\).
complete the proof by selecting the correct reasoning for steps 3 and 6.
complete the proof.

statements	reasons
2. \\( \overleftrightarrow { e f } \\) is the transversal of \\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c d } \\).	2. definition of a transversal
3. \\( \angle h k d \cong \angle a h k \\)	3.
4. \\( \angle a h k \cong \angle e h b \\)	4.
5. \\( \angle e h b \cong \angle c k f \\)	5.
6. \\( \angle c k f \cong \angle h k d \\)	6.
7. \\( \angle h k d \cong \angle e h b \\)	7. transitive property of congruence

Explanation:

Step1: Reason for statement 1

The first statement is given, so the reason is "Given".

Step2: Reason for statement 3

Since \( \overleftrightarrow{AB}\parallel\overleftrightarrow{CD}\) and \( \overleftrightarrow{EF}\) is the transversal, \( \angle HKD\) and \( \angle AHK\) are alternate - interior angles. By the Alternate - Interior Angles Theorem, if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. So the reason is "Alternate interior angles are congruent".

Step3: Reason for statement 6

\( \angle CKF\) and \( \angle HKD\) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason is "Vertical angles are congruent".

Answer:

1. Given; 3. Alternate interior angles are congruent; 6. Vertical angles are congruent.