Question

in the figure, lines hj and kl are parallel. match each angle relationship with the appropriate reason. vertical angles theorem, same - side interior angles, corresponding angles theorem, alternate interior angles theorem, alternate exterior angles theorem, linear pair. m∠lfg + m∠gfk = 180°, m∠hed = m∠fej, m∠dej = m∠efl, m∠kfe = m∠fej

Explanation:

Step1: Analyze \(m\angle LFG + m\angle GFK=180^{\circ}\)

These angles form a straight - line. By definition, angles that form a straight - line are a linear pair.

Step2: Analyze \(m\angle HED = m\angle FEJ\)

\(\angle HED\) and \(\angle FEJ\) are opposite each other when two lines intersect. They are vertical angles, so we use the vertical angles theorem.

Step3: Analyze \(m\angle DEJ = m\angle EFL\)

\(\angle DEJ\) and \(\angle EFL\) are in the same relative position with respect to the parallel lines \(HJ\) and \(KL\) and the transversal \(DG\). They are corresponding angles, so we use the corresponding angles theorem.

Step4: Analyze \(m\angle KFE = m\angle FEJ\)

\(\angle KFE\) and \(\angle FEJ\) are between the two parallel lines \(HJ\) and \(KL\) and on opposite sides of the transversal \(DG\). They are alternate interior angles, so we use the alternate interior angles theorem.

Answer:

\(m\angle LFG + m\angle GFK = 180^{\circ}\to\) linear pair
\(m\angle HED = m\angle FEJ\to\) vertical angles theorem
\(m\angle DEJ = m\angle EFL\to\) corresponding angles theorem
\(m\angle KFE = m\angle FEJ\to\) alternate interior angles theorem