Let $M_1$ be the molar mass of $AB_2$ and $M_2$ be the molar mass of $AB$.

We are given that $\Delta T_f = K_f \cdot m$, where $m$ is the molality.

For $AB_2$: $\Delta T_{f1} = 0.689 = K_f \cdot \frac{1/M_1}{50/1000} = K_f \cdot \frac{1000}{50M_1} = K_f \cdot \frac{20}{M_1}$

For $AB$: $\Delta T_{f2} = 1.176 = K_f \cdot \frac{1/M_2}{50/1000} = K_f \cdot \frac{1000}{50M_2} = K_f \cdot \frac{20}{M_2}$

Dividing the two equations, we get:

$\frac{0.689}{1.176} = \frac{K_f \cdot \frac{20}{M_1}}{K_f \cdot \frac{20}{M_2}} = \frac{M_2}{M_1}$

Therefore, $M_2 = M_1 \cdot \frac{0.689}{1.176} \approx 0.5858 M_1$

We also know that $M_1 = M_A + 2M_B$ and $M_2 = M_A + M_B$.

So, $M_A + M_B = 0.5858(M_A + 2M_B)$

$M_A + M_B = 0.5858M_A + 1.1716M_B$

$0.4142M_A = 0.1716M_B$

$M_A = \frac{0.1716}{0.4142}M_B \approx 0.4143 M_B$

Substituting $M_A$ in $M_1$:

$M_1 = M_A + 2M_B = 0.4143M_B + 2M_B = 2.4143M_B$

Substituting $M_A$ in $M_2$:

$M_2 = M_A + M_B = 0.4143M_B + M_B = 1.4143M_B$

From $\Delta T_{f2} = 1.176 = K_f \cdot \frac{20}{M_2}$, we have $K_f = \frac{1.176 M_2}{20} = \frac{1.176 \times 1.4143 M_B}{20} = 0.0831 M_B$

From $\Delta T_{f1} = 0.689 = K_f \cdot \frac{20}{M_1}$, we have $K_f = \frac{0.689 M_1}{20} = \frac{0.689 \times 2.4143 M_B}{20} = 0.0831 M_B$

Using $\Delta T_{f1} = 0.689 = K_f \cdot \frac{20}{M_1}$ and $\Delta T_{f2} = 1.176 = K_f \cdot \frac{20}{M_2}$

We have $\frac{0.689}{20} M_1 = \frac{1.176}{20} M_2 = K_f$

Therefore, $0.689 M_1 = 1.176 M_2$

Also, $M_2 = M_A + M_B$ and $M_1 = M_A + 2M_B$

So, $0.689(M_A + 2M_B) = 1.176(M_A + M_B)$

$0.689M_A + 1.378M_B = 1.176M_A + 1.176M_B$

$0.487M_A = 0.202M_B$

$M_A = \frac{0.202}{0.487} M_B = 0.4148 M_B$

Then $M_1 = M_A + 2M_B = 0.4148 M_B + 2M_B = 2.4148 M_B$

And $M_2 = M_A + M_B = 0.4148 M_B + M_B = 1.4148 M_B$

From $\Delta T_{f1} = 0.689 = K_f \cdot \frac{1}{M_1} \cdot \frac{1000}{50}$, we get $0.689 = K_f \cdot \frac{20}{M_1}$

From $\Delta T_{f2} = 1.176 = K_f \cdot \frac{1}{M_2} \cdot \frac{1000}{50}$, we get $1.176 = K_f \cdot \frac{20}{M_2}$

Dividing, $\frac{0.689}{1.176} = \frac{M_2}{M_1} = \frac{1.4148 M_B}{2.4148 M_B} = 0.5859$

So, $M_2 = 0.5859 M_1$

Substituting in $1.176 = K_f \cdot \frac{20}{M_2}$, we get $1.176 = K_f \cdot \frac{20}{0.5859 M_1}$

$K_f = \frac{1.176 \times 0.5859 M_1}{20} = 0.0344 M_1$

Substituting in $0.689 = K_f \cdot \frac{20}{M_1}$, we get $0.689 = 0.0344 M_1 \cdot \frac{20}{M_1} = 0.688$

This doesn't help us find $M_1$.

Let's try a different approach.

$\frac{\Delta T_{f1}}{\Delta T_{f2}} = \frac{0.689}{1.176} = \frac{M_2}{M_1}$

$M_2 = \frac{0.689}{1.176} M_1 = 0.5858 M_1$

We have $M_1 = M_A + 2M_B$ and $M_2 = M_A + M_B$.

So $M_A + M_B = 0.5858(M_A + 2M_B)$

$M_A + M_B = 0.5858 M_A + 1.1716 M_B$

$0.4142 M_A = 0.1716 M_B$

$M_A = \frac{0.1716}{0.4142} M_B = 0.4143 M_B$

Now, $\Delta T_{f1} = K_f \cdot \frac{1}{M_1} \cdot \frac{1000}{50} = K_f \cdot \frac{20}{M_1} = 0.689$

$\Delta T_{f2} = K_f \cdot \frac{1}{M_2} \cdot \frac{1000}{50} = K_f \cdot \frac{20}{M_2} = 1.176$

We have two equations and three unknowns ($K_f, M_1, M_2$).

We also have $M_2 = 0.5858 M_1$.

Let's assume $M_1 = 208$. Then $M_2 = 0.5858 \times 208 = 121.8464 \approx 122$

Then $K_f = \frac{0.689 \times 208}{20} = 7.1656$ and $K_f = \frac{1.176 \times 122}{20} = 7.1712$

So $M_1 \approx 208$ is a good estimate.