Question

Given: NL is a diagonal of parallelogram KLMN.

Prove: KL ≅ NM and KN ≅ LM

A parallelogram K L M N is divided into two triangles. N L is the diagonal of the parallelogram.

Proof:
Statements Reasons
1. NL is a diagonal of parallelogram KLMN. Given
2. KL ∥ NM and KN ∥ LM Definition of a parallelogram
3. ? ?
4. LN ≅ NL Reflexive property of congruence
5. ΔKLN ≅ ΔMNL ASA congruence criteria
6. KL ≅ NM and KN ≅ LM Corresponding parts of congruent triangles are congruent

Select the missing statement and reason to complete the given proof.
A.
∠LNK ≅ ∠NLM, and ∠KLN ≅ ∠MNL
by the corresponding angles theorem
B.
∠KLN ≅ ∠MLN, and ∠KNL ≅ ∠MNL
by the alternate interior angles theorem
C.
∠KLN ≅ ∠MLN, and ∠KNL ≅ ∠MNL
by the corresponding angles theorem
D.
∠LNK ≅ ∠NLM, and ∠KLN ≅ ∠MNL
by the alternate interior angles theorem

Answer 1

To complete the given proof, we need to select the missing statement and reason that correctly demonstrates the congruence of angles in the triangles ΔKLN and ΔMNL.

The correct option is:

C.
`\displaystyle\sf \angle KLN \cong \angle MLN`, and
`\displaystyle\sf \angle KNL \cong \angle MNL`

by the corresponding angles theorem

The corresponding angles theorem states that if two parallel lines are intersected by a transversal, then the corresponding angles formed are congruent. In this case,
`\displaystyle\sf KL \parallel NM` and
`\displaystyle\sf KN \parallel LM`, so the corresponding angles
`\displaystyle\sf \angle KLN` and
`\displaystyle\sf \angle MLN`, as well as
`\displaystyle\sf \angle KNL` and
`\displaystyle\sf \angle MNL`, are congruent. Therefore, option C is the correct choice to complete the proof.

Answer 2

The missing statement is C. ∠KLN ≅ ∠MLN, and ∠KNL ≅ ∠MNL by the corresponding angles theorem.

Step-by-step explanation:

In the given proof, the missing statement needs to establish the congruence of corresponding angles in triangles ΔKLN and ΔMNL. The corresponding angles theorem states that if a transversal intersects two parallel lines, then the corresponding angles are congruent. In this case, KL ∥ NM and KN ∥ LM by the definition of a parallelogram.

Statement 3 should establish that ∠KLN ≅ ∠MLN and ∠KNL ≅ ∠MNL. Option C correctly states this by applying the corresponding angles theorem, ensuring that the corresponding angles in the triangles formed by the parallelogram's diagonals are congruent.

To elaborate, in triangles ΔKLN and ΔMNL, ∠KLN corresponds to ∠MLN, and ∠KNL corresponds to ∠MNL. This congruence is a crucial step in proving triangles congruent by the ASA (angle-side-angle) congruence criteria. With ∠KLN ≅ ∠MLN and ∠KNL ≅ ∠MNL established, the proof proceeds to use ASA to conclude that ΔKLN ≅ ΔMNL. This congruence allows us to then deduce that KL ≅ NM and KN ≅ LM by the corresponding parts of congruent triangles.