Given: ∠ALN ≅ ∠DNL,∠ANL≅∠DLN

Question

Prove: LAND is a parallelogram.

Quadrilateral LAND has diagonal LN, with angles ALN and DNL congruent and angles ANL and DLN congruent.

(1 point)
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If ∠ALN ≅ ∠DNL, then || .

If ∠ANL≅∠DLN, then || .

Question 2
What theorem was used to identify that the opposite sides are parallel?(1 point)
Responses

If the same side interior angles are supplementary, then the lines are parallel
If the same side interior angles are supplementary, then the lines are parallel

If the alternate exterior angles are congruent, then the lines are parallel
If the alternate exterior angles are congruent, then the lines are parallel

If the alternate interior angles are congruent, then the lines are parallel
If the alternate interior angles are congruent, then the lines are parallel

If the corresponding angles are congruent, then the lines are parallel
If the corresponding angles are congruent, then the lines are parallel
Question 3
Which of the following ways was used in this problem to confirm that the quadrilateral was a parallelogram?(1 point)
Responses

If the diagonals bisect each other, then the quadrilateral is a parallelogram
If the diagonals bisect each other, then the quadrilateral is a parallelogram

If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram
If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram

If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram
If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram

If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram
If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram

If both pairs of opposite angles are congruent, then the quadrilateral is a

Answer 1

Let's fill out the responses step by step based on the given information about quadrilateral LAND and the properties of the angles:

Question 1

Given:

• \( \angle ALN \cong \angle DNL \)
• \( \angle ANL \cong \angle DLN \)

To prove that angles imply parallel sides:

1. If \( \angle ALN \cong \angle DNL \), then \( LA \parallel ND \).
2. If \( \angle ANL \cong \angle DLN \), then \( LN \parallel AD \).

Question 2

The theorem used to identify that the opposite sides are parallel is:

• If the alternate interior angles are congruent, then the lines are parallel.

Question 3

The way used to confirm that the quadrilateral was a parallelogram:

• If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram.

Putting it all together, here are the final answers:

1. If \( \angle ALN \cong \angle DNL \), then \( LA \parallel ND \).
2. If \( \angle ANL \cong \angle DLN \), then \( LN \parallel AD \).
3. If the alternate interior angles are congruent, then the lines are parallel.
4. If one pair of sides is both congruent and parallel, then the quadrilateral is a parallelogram.