Question

the figure shows square klmn. which of the following conditions satisfies the criteria for squares? lk ≅ ln, lm ≅ km, lm || lk, lm || kn

Explanation:

Step1: Recall square properties

In a square, all sides are equal, diagonals are equal and bisect each other at 90 degrees, and opposite sides are parallel. Also, in a square, the diagonals are equal in length. Let's analyze each option:

- Option 1: \( \overline{LK} \cong \overline{LN} \): \( LK \) is a side, \( LN \) is a diagonal. In a square, diagonal \( = \text{side} \times \sqrt{2} \), so \( LK eq LN \) (unless side is 0, which it's not). So this is false.
- Option 2: \( \overline{LM} \cong \overline{KM} \): \( LM \) is a side, \( KM \) is a diagonal. Similar to above, diagonal \( = \text{side} \times \sqrt{2} \), so \( LM eq KM \) (unless side is 0). Wait, no—wait, in square KLMN, vertices are K, L, M, N in order. So sides: KL, LM, MN, NK. Diagonals: KM and LN. Wait, maybe I mislabeled. Wait, square KLMN: so K to L to M to N to K. So sides: KL, LM, MN, NK. Diagonals: KM (from K to M) and LN (from L to N). In a square, diagonals are equal, so KM = LN. But LM is a side, KM is a diagonal. Wait, no—wait, maybe the labels: Let's see the figure: L and M are top, K and N are bottom. So KL is left side, LM is top side, MN is right side, NK is bottom side. Diagonals: KM (from K (bottom left) to M (top right)) and LN (from L (top left) to N (bottom right)). So in a square, all sides are equal, diagonals are equal. Now, LM is a side, KM is a diagonal. Wait, no—wait, maybe the option is \( \overline{LM} \cong \overline{KM} \)? No, that can't be. Wait, maybe I made a mistake. Wait, no—wait, the other options: \( LM \parallel LK \)? No, LM and LK are adjacent sides, so they are perpendicular, not parallel. \( LM \parallel KN \): LM is top side, KN is bottom side. In a square, opposite sides are parallel, so LM (top) and KN (bottom) are parallel. Wait, but let's re-examine the options. Wait, the options are:

1. \( \overline{LK} \cong \overline{LN} \): LK is left side, LN is diagonal (from L to N, bottom right). So LK is side, LN is diagonal: length of diagonal is \( \text{side} \times \sqrt{2} \), so not congruent.

2. \( \overline{LM} \cong \overline{KM} \): LM is top side, KM is diagonal (from K to M). Diagonal length is \( \text{side} \times \sqrt{2} \), so LM (side) and KM (diagonal) are not congruent.

3. \( \overline{LM} \parallel \overline{LK} \): LM is top side (horizontal), LK is left side (vertical). They are perpendicular, not parallel.

4. \( \overline{LM} \parallel \overline{KN} \): LM is top side (horizontal), KN is bottom side (horizontal). In a square, opposite sides are parallel, so LM and KN are parallel. Wait, but that's a property of rectangles (and squares) that opposite sides are parallel. But the question is about the criteria for squares. Wait, but maybe the options are miswritten? Wait, no—wait, maybe the first option: \( \overline{LK} \cong \overline{LN} \) is wrong, second: \( \overline{LM} \cong \overline{KM} \) is wrong, third: \( LM \parallel LK \) is wrong (perpendicular), fourth: \( LM \parallel KN \) is correct because in a square (and rectangle), opposite sides are parallel. But wait, maybe the correct answer is \( \overline{LM} \parallel \overline{KN} \). Wait, but let's check again. Wait, the square has sides: KL, LM, MN, NK. So KL is left, LM is top, MN is right, NK is bottom. So LM (top) and NK (bottom) are opposite sides, so they are parallel. So \( LM \parallel KN \) (since KN is same as NK, just labeled from K to N, which is bottom side). So that's a property of parallelograms (and squares, rectangles, rhombuses) that opposite sides are parallel. But the question is about the criteria for squares. Wait, maybe the options are different. Wait, maybe I misread the options. Let's re-express:

Options:

1. \( \overline{LK} \cong \overline{LN} \): LK is side, LN is diagonal. Not congruent.

2. \( \overline{LM} \cong \overline{KM} \): LM is side, KM is diagonal. Not congruent.

3. \( \overline{LM} \parallel \overline{LK} \): LM (top) and LK (left) are adjacent, perpendicular. Not parallel.

4. \( \overline{LM} \parallel \overline{KN} \): LM (top) and KN (bottom) are opposite sides, parallel. So this is true for squares (and rectangles, parallelograms). But maybe the question is about the criteria, but among the options, this is the only correct one. Wait, but maybe the first option: \( \overline{LK} \cong \overline{LN} \) is wrong, second: \( \overline{LM} \cong \overline{KM} \) is wrong, third: parallel between adjacent sides is wrong, fourth: parallel between opposite sides is correct. So the correct option is the fourth one: \( \overline{LM} \parallel \overline{KN} \).

Wait, but maybe I made a mistake. Let's confirm: In a square, opposite sides are parallel. So LM (top) and KN (bottom) are parallel. So that's a valid property. So the correct condition is \( \overline{LM} \parallel \overline{KN} \).

Answer:

The correct option is the one with \( \overline{LM} \parallel \overline{KN} \) (the last option, assuming the options are listed as: \( \overline{LK} \cong \overline{LN} \), \( \overline{LM} \cong \overline{KM} \), \( \overline{LM} \parallel \overline{LK} \), \( \overline{LM} \parallel \overline{KN} \)). So the answer is the option with \( \overline{LM} \parallel \overline{KN} \).