Understanding Parallelogram Angles

In a parallelogram, consecutive (adjacent) angles add up to $180^\circ$, and opposite angles are equal. Let the smallest angle be represented by '$x$'.

Setting Up the Angle Equation

According to the problem, one angle is '$30^\circ$' less than twice the smallest angle. This other angle can be expressed as '$2x - 30^\circ$'.

Since these must be adjacent angles in the parallelogram, their sum is $180^\circ$: $x + (2x - 30^\circ) = 180^\circ$

Solving for the Smallest Angle

Now, we solve the equation for '$x$':

1. Combine like terms: $3x - 30^\circ = 180^\circ$
2. Add $30^\circ$ to both sides: $3x = 180^\circ + 30^\circ$ $3x = 210^\circ$
3. Divide by 3 to find '$x$': $x = \frac{210^\circ}{3}$ $x = 70^\circ$

So, the smallest angle ($x$) is $70^\circ$. The angles in the parallelogram are $70^\circ$ and $180^\circ - 70^\circ = 110^\circ$.

Determining the Largest Angle

The two distinct angle measures in the parallelogram are $70^\circ$ (the smallest) and $110^\circ$. We can verify the condition: $2 \times 70^\circ - 30^\circ = 140^\circ - 30^\circ = 110^\circ$. This confirms our angle values.

The largest angle is therefore $110^\circ$.

Final Answer

The measure of the largest angle of the parallelogram is $110^\circ$.