Question

1. given: \\( \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } \\) and \\( \overrightarrow { 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 1. \\( \overleftrightarrow { a b } \parallel \overleftrightarrow { c d } \\) 1. 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 step 3

When two parallel lines ($\overleftrightarrow{AB}\parallel\overleftrightarrow{CD}$) are cut by a transversal ($\overleftrightarrow{EF}$), the alternate - interior angles are congruent. $\angle HKD$ and $\angle AHK$ are alternate - interior angles.

Step2: Reason for step 6

Vertical angles are congruent. $\angle CKF$ and $\angle HKD$ are vertical angles.

Answer:

1. Alternate - interior angles theorem; 6. Vertical angles theorem