Question

If one angle of a parallelogram is $24^\circ$ less than twice the smallest angle, then the measure of the largest angle of the parallelogram is :

• A. $176^\circ$
• B. $68^\circ$
• C. $112^\circ$
• D. $102^\circ$

Hint:

Always remember the fundamental properties of parallelograms: consecutive angles are supplementary, and opposite angles are equal. This allows you to set up equations to solve for unknown angles.

Answer 1

The Correct Option is C

Step 1: Recall properties of a parallelogram.
In a parallelogram, consecutive angles are supplementary (sum to $180^\circ$), and opposite angles are equal.
Let the two distinct angles of the parallelogram be $\alpha$ and $\beta$. We know that $\alpha + \beta = 180^\circ$.
Step 2: Set up equations based on the problem statement.
Let the smallest angle be $S$.
Let the other angle be $L$.
From the problem statement, "one angle of a parallelogram is $24^\circ$ less than twice the smallest angle".
So, $L = 2S - 24^\circ$.
Also, we know that the sum of adjacent angles in a parallelogram is $180^\circ$:
$S + L = 180^\circ$ Step 3: Solve the system of equations for S and L.
Substitute the expression for $L$ into the sum equation:
$S + (2S - 24^\circ) = 180^\circ$
$3S - 24^\circ = 180^\circ$
$3S = 180^\circ + 24^\circ$
$3S = 204^\circ$
$S = \frac{204^\circ}{3}$
$S = 68^\circ$
Now, find $L$:
$L = 2S - 24^\circ$
$L = 2(68^\circ) - 24^\circ$
$L = 136^\circ - 24^\circ$
$L = 112^\circ$
Step 4: Identify the largest angle.
The two angles are $S = 68^\circ$ and $L = 112^\circ$.
The largest angle of the parallelogram is $112^\circ$.
(Check: $68^\circ + 112^\circ = 180^\circ$, which is correct for adjacent angles of a parallelogram).
Step 5: Compare with the given options.
The calculated largest angle is $112^\circ$, which matches option (3). (3) 112\textdegree