Question

complete the proof.

1. ( overleftrightarrow{ab} parallel overleftrightarrow{cd} )
2. ( overleftrightarrow{ef} ) is the transversal of ( overleftrightarrow{ab} ) and ( overleftrightarrow{cd} ).
3. ( angle hkd cong angle ahk )
4. ( angle ahk cong angle ehb )
5. ( angle ehb cong angle ckf )
6. ( angle ckf cong angle hkd )
7. ( angle hkd cong angle ehb )

Explanation:

Step1: Reason for Statement 1

Given information.

Step2: Reason for Statement 3

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

Step3: Reason for Statement 4

Vertical angles are congruent. \(\angle AHK\) and \(\angle EHB\) are vertical angles.

Step4: Reason for Statement 5

Vertical angles are congruent. \(\angle EHB\) and \(\angle CKF\) are vertical angles.

Step5: Reason for Statement 6

Transitive property of congruence. If \(\angle CKF\cong\angle EHB\) (from step 5) and \(\angle EHB\cong\angle AHK\) (from step 4) and \(\angle AHK\cong\angle HKD\) (from step 3), then \(\angle CKF\cong\angle HKD\).

Answer:

1. Given
2. Alternate - interior angles theorem
3. Vertical angles theorem
4. Vertical angles theorem
5. Transitive property of congruence