Question

given: $ef \parallel gh$ and $ab \parallel cd$.
prove: $\angle ijl$ and $\angle jlk$ are supplementary.
note: quadrilateral properties are not permitted in this proof.
step 1
statement: $\overline{ef} \parallel \overline{gh}$, $\overline{ab} \parallel \overline{cd}$
reason: given
(there is a dropdown menu labeled type of statement and a diagram with lines $ef$, $gh$, $ab$, $cd$ and points $i$, $j$, $k$, $l$ as shown.)

Explanation:

Step 1: Identify Corresponding Angles (EF || GH)

Since \( \overline{EF} \parallel \overline{GH} \) and \( \overline{AB} \) is a transversal, \( \angle IJL \) and \( \angle IKL \) are corresponding angles. By the Corresponding Angles Postulate, \( \angle IJL \cong \angle IKL \).

Step 2: Identify Supplementary Angles (AB || CD)

Since \( \overline{AB} \parallel \overline{CD} \) and \( \overline{EF} \) is a transversal, \( \angle IKL \) and \( \angle JLK \) are same - side interior angles. By the Same - Side Interior Angles Theorem, \( \angle IKL+\angle JLK = 180^{\circ}\), meaning \( \angle IKL \) and \( \angle JLK \) are supplementary.

Step 3: Substitute \( \angle IJL \) for \( \angle IKL \)

Since \( \angle IJL \cong \angle IKL \) (from Step 1), we can substitute \( \angle IJL \) for \( \angle IKL \) in the equation \( \angle IKL+\angle JLK = 180^{\circ} \). So, \( \angle IJL+\angle JLK = 180^{\circ} \), which means \( \angle IJL \) and \( \angle JLK \) are supplementary.

Answer:

\( \angle IJL \) and \( \angle JLK \) are supplementary (proven as above).