Select the missing statement and reason to complete the given proof. A. $$ \angle L N K \cong \angle N L M $$, and $$ \angle K L N \cong \angle M N L $$ by the corresponding angles theorem B. $$ \angle K L N \cong \angle M L N $$, and $$ \angle K N L \cong \angle M N L $$ by the alternate interior angles theorem C. $$ \angle L N K \cong \angle N L M $$, and $$ \angle K L N \cong \angle M N L $$ by the alternate interior angles theorem D. $$ \begin{array}{l}\angle K L N \cong \angle M L N, ext { and } \angle K N L \cong \angle M N L \ ext { by the corresponding angles theorem }\end{array} $$

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Select the missing statement and reason to complete the given proof. A. $$ \angle L N K \cong \angle N L M $$, and $$ \angle K L N \cong \angle M N L $$ by the corresponding angles theorem B. $$ \angle K L N \cong \angle M L N $$, and $$ \angle K N L \cong \angle M N L $$ by the alternate interior angles theorem C. $$ \angle L N K \cong \angle N L M $$, and $$ \angle K L N \cong \angle M N L $$ by the alternate interior angles theorem D. $$ \begin{array}{l}\angle K L N \cong \angle M L N, 	ext { and } \angle K N L \cong \angle M N L \ 	ext { by the corresponding angles theorem }\end{array} $$

UpStudy AI · Accepted Answer

To determine the correct statement and reason for completing the proof, let's analyze each option based on the properties of corresponding angles and alternate interior angles.

**Option D:**
$$
\begin{array}{l}
\angle K L N \cong \angle M L N, \text{ and } \angle K N L \cong \angle M N L \
\text{ by the corresponding angles theorem }
\end{array}
$$

- **$$\angle K L N \cong \angle M L N$$:** These angles are formed when a transversal intersects two parallel lines. They are in corresponding positions, making them congruent by the corresponding angles theorem.
  
- **$$\angle K N L \cong \angle M N L$$:** Similarly, these angles occupy corresponding positions relative to the transversal and the parallel lines, ensuring their congruence by the corresponding angles theorem.

**Why the Other Options Are Incorrect:**

- **Option A:** While the angle congruences are correct, they are actually alternate interior angles, not corresponding angles.
  
- **Option B:** This option incorrectly attributes the angle congruences to the alternate interior angles theorem when they are actually corresponding angles.
  
- **Option C:** Similar to Option A, it correctly identifies the angle congruences but mislabels them as alternate interior angles instead of corresponding angles.

**Conclusion:**
Option **D** correctly identifies both the congruent angles and the appropriate theorem used to establish their congruence.

**Answer:**  
**D.**  
$$
\begin{array}{l}
\angle K L N \cong \angle M L N, \text{ and } \angle K N L \cong \angle M N L \
\text{ by the corresponding angles theorem }
\end{array}
$$
**Answer:** D. $$ \begin{array}{l} \angle K L N \cong \angle M L N, \text{ and } \angle K N L \cong \angle M N L \ \text{ by the corresponding angles theorem } \end{array} $$

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