Figure 7. Given parallelogram LMCF, with LM = MF and ∠MLF = 90°. Prove that LMCF is a square.

Proof:

We are given a parallelogram LMCF with the following properties:

1. LMCF is a parallelogram.
2. LM = MF.
3. ∠MLF = 90°.

We need to prove that LMCF is a square.

A square is a quadrilateral that has four equal sides and four right angles. Alternatively, a square is a rectangle with all sides equal, or a rhombus with one right angle.

Let's use the properties of a parallelogram and the given information:

1. Property of parallelogram: Opposite sides are equal in length. So, LM = CF and MC = LF.
2. Property of parallelogram: Opposite angles are equal. So, ∠LMC = ∠LFC and ∠MLF = ∠MCF.
3. We are given LM = MF.
4. From property 1, since LM = CF, we have CF = MF.
5. Also, from property 1, since MC = LF.
6. Now consider sides LM and MF. We are given LM = MF.
7. Since LMCF is a parallelogram, opposite sides are equal: LM = CF and MF = LC. Therefore, LM = MF = CF = LC. All four sides are equal. This means LMCF is a rhombus.
8. We are given that ∠MLF = 90°.
9. Since LMCF is a parallelogram, opposite angles are equal, so ∠MCF = ∠MLF = 90°.
10. Adjacent angles in a parallelogram are supplementary. So, ∠LMC + ∠MLF = 180°.
11. ∠LMC + 90° = 180°.
12. ∠LMC = 180° - 90° = 90°.
13. Since opposite angles are equal, ∠LFC = ∠LMC = 90°.

So, we have shown that LMCF has four equal sides (it's a rhombus) and four right angles.

Conclusion: Since LMCF is a parallelogram with four equal sides and one right angle (which implies all angles are right angles), it is a square.