Solving the Parallelogram Angle Problem

Let the smallest angle of the parallelogram be denoted by $x$. According to the problem statement, one angle is $39^\circ$ less than twice the smallest angle. This other angle can be represented as $2x - 39^\circ$.

In a parallelogram, adjacent angles are supplementary, meaning they add up to $180^\circ$. Therefore, we can set up an equation relating the smallest angle and the other angle:

$x + (2x - 39^\circ) = 180^\circ$

Calculating the Smallest Angle

Now, we solve the equation for $x$:

• Combine like terms: $3x - 39^\circ = 180^\circ$
• Add $39^\circ$ to both sides: $3x = 180^\circ + 39^\circ$ $3x = 219^\circ$
• Divide by 3 to find the value of $x$: $x = \frac{219^\circ}{3}$ $x = 73^\circ$

The smallest angle is $73^\circ$. We can verify this: the other angle is $2(73^\circ) - 39^\circ = 146^\circ - 39^\circ = 107^\circ$. Since $73^\circ + 107^\circ = 180^\circ$, the angles are correct for a parallelogram, and $73^\circ$ is indeed the smallest angle.