Quadrilateral Angles Ratio Explained

The problem asks for the difference between the largest and smallest angles of a quadrilateral, given that the angles are in the ratio 1 : 3 : 4 : 7.

Finding the Angles of the Quadrilateral

Let the angles of the quadrilateral be $1x$, $3x$, $4x$, and $7x$, based on the given ratio.

The sum of the interior angles of any quadrilateral is always $360^\circ$. Therefore, we can set up the equation:

$1x + 3x + 4x + 7x = 360^\circ$

Combine the terms:

$15x = 360^\circ$

Solve for $x$ by dividing both sides by 15:

$x = \frac{360^\circ}{15}$

$x = 24^\circ$

Calculating Angle Values

Now, we can find the measure of each angle:

• Smallest angle: $1x = 1 \times 24^\circ = 24^\circ$
• Second angle: $3x = 3 \times 24^\circ = 72^\circ$
• Third angle: $4x = 4 \times 24^\circ = 96^\circ$
• Largest angle: $7x = 7 \times 24^\circ = 168^\circ$

Determining the Angle Difference

The question asks for the difference between the largest and smallest angles.

• Largest angle = $168^\circ$
• Smallest angle = $24^\circ$

Difference = Largest angle $-$ Smallest angle

Difference = $168^\circ - 24^\circ$

Difference = $144^\circ$

Conclusion

The difference between the largest and smallest angles of the quadrilateral is $144^\circ$. This corresponds to Option D.