Question

given: \\(\overline{ef} \parallel \overline{gh}\\) and \\(\overline{ab} \parallel \overline{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 drop - down box labeled type of statement below the statement)
(there is a diagram at the bottom: there are two horizontal lines \\(ef\\) and \\(gh\\) (\\(ef\\) is above \\(gh\\)), and two oblique lines \\(ab\\) and \\(cd\\) (\\(ab\\) is on the left of \\(cd\\)). the intersection of \\(ab\\) and \\(ef\\) is \\(i\\), the intersection of \\(ab\\) and \\(gh\\) is \\(j\\); the intersection of \\(cd\\) and \\(ef\\) is \\(k\\), and the intersection of \\(cd\\) and \\(gh\\) is \\(l\\).)

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 Same-Side Interior 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, $\angle IKL+\angle JLK = 180^{\circ}$ (supplementary angles).

Step 3: Substitute $\angle IJL$ for $\angle IKL$

From Step 1, we know that $\angle IJL\cong\angle IKL$, which means $\angle IJL=\angle IKL$. Substituting $\angle IJL$ for $\angle IKL$ in the equation $\angle IKL+\angle JLK = 180^{\circ}$, we get $\angle IJL+\angle JLK=180^{\circ}$. So, $\angle IJL$ and $\angle JLK$ are supplementary.

Answer:

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