The problem states that three interior angles of a quadrilateral are $75^{\circ}$, $105^{\circ}$, and $100^{\circ}$. It asks whether a circle can be circumscribed around the quadrilateral.

1. Problem Description

The problem states that three interior angles of a quadrilateral are $75^{\circ}$, $105^{\circ}$, and $100^{\circ}$. It asks whether a circle can be circumscribed around the quadrilateral.

2. Solution Steps

A quadrilateral can be circumscribed by a circle if and only if the sums of its opposite angles are equal to $180^{\circ}$.

First, we need to find the fourth angle of the quadrilateral.

The sum of the interior angles of a quadrilateral is $360^{\circ}$. Let the fourth angle be $x$. Then we have:

$75^{\circ} + 105^{\circ} + 100^{\circ} + x = 360^{\circ}$

$280^{\circ} + x = 360^{\circ}$

$x = 360^{\circ} - 280^{\circ}$

$x = 80^{\circ}$

Now we have the four angles of the quadrilateral: $75^{\circ}$, $105^{\circ}$, $100^{\circ}$, and $80^{\circ}$.

We need to check if the sums of opposite angles are equal to $180^{\circ}$.

Case 1: $75^{\circ} + 100^{\circ} = 175^{\circ} \neq 180^{\circ}$

$105^{\circ} + 80^{\circ} = 185^{\circ} \neq 180^{\circ}$

Case 2: $75^{\circ} + 80^{\circ} = 155^{\circ} \neq 180^{\circ}$

$105^{\circ} + 100^{\circ} = 205^{\circ} \neq 180^{\circ}$

Case 3: $75^{\circ} + 105^{\circ} = 180^{\circ}$

$100^{\circ} + 80^{\circ} = 180^{\circ}$

Since the sums of one pair of opposite angles equal $180^{\circ}$, a circle can be circumscribed around the quadrilateral.

3. Final Answer

Yes, a circle can be circumscribed around the quadrilateral.