Question

a diagram is shown where (overleftrightarrow{fp}) intersects (overleftrightarrow{rt}), (overleftrightarrow{hd}), and (overleftrightarrow{zk}).
given: (rtparallel zk), (angle tmvcongangle dva)
prove: (hdparallel zk)

statement	reason
2. (angle tmvcongangle pak)	select
3. (angle tmvcongangle dva)	3. given
4. (angle pakcongangle dva)	select
5. (overleftrightarrow{hd}paralleloverleftrightarrow{zk})	select

Explanation:

Step1: Identify the property for \(\angle TMV\cong\angle PAK\)

Since \(RT\parallel ZK\), by the Corresponding Angles Theorem, \(\angle TMV\cong\angle PAK\) (when two parallel lines are cut by a transversal, corresponding angles are congruent).

Step2: Justify \(\angle PAK\cong\angle DVA\)

We know that \(\angle TMV\cong\angle PAK\) (from Step 1) and \(\angle TMV\cong\angle DVA\) (given). By the Transitive Property of Congruence (\(a = b\) and \(b = c\) implies \(a = c\) for congruent angles), \(\angle PAK\cong\angle DVA\).

Step3: Prove \(HD\parallel ZK\)

Since \(\angle PAK\cong\angle DVA\), by the Converse of the Corresponding Angles Theorem (if two lines are cut by a transversal such that corresponding angles are congruent, then the two lines are parallel), \(HD\parallel ZK\).

Answer:

1. Corresponding Angles Theorem; 4. Transitive Property of Congruence; 5. Converse of the Corresponding Angles Theorem.