Parallelogram Angle Calculation

This solution explains how to find the largest angle of a parallelogram using the properties of its angles and basic algebra.

Problem Setup

• Let the smallest angle of the parallelogram be denoted by $x$.
• In any parallelogram, adjacent angles sum to $180^\circ$. Therefore, the other distinct angle in the parallelogram is $180^\circ - x$.
• The angles of the parallelogram are $x$, $180^\circ - x$, $x$, and $180^\circ - x$.
• The problem states that one angle is $18^\circ$ less than twice the smallest angle.

Equation Formulation

• Let the smallest angle be $x$.
• The given condition translates to: One angle $= 2x - 18^\circ$.
• Since $x$ is the smallest angle, the angle $180^\circ - x$ represents the larger angle. This larger angle must satisfy the given condition.
• Set up the equation: $180^\circ - x = 2x - 18^\circ$

Solving for the Smallest Angle

• To find the value of $x$, rearrange the equation:
$180^\circ + 18^\circ = 2x + x$ $198^\circ = 3x$
• Divide by 3 to solve for $x$:
$x = \frac{198^\circ}{3}$ $x = 66^\circ$
• Thus, the smallest angle of the parallelogram is $66^\circ$.

Calculating the Largest Angle

• The largest angle is the supplementary angle to the smallest angle, calculated as $180^\circ - x$.
$\text{Largest Angle} = 180^\circ - 66^\circ$ $\text{Largest Angle} = 114^\circ$
• Verification: Twice the smallest angle is $2 \times 66^\circ = 132^\circ$. Subtracting $18^\circ$ gives $132^\circ - 18^\circ = 114^\circ$, confirming our result.

The largest angle of the parallelogram is $114^\circ$.