Final answer:
A circle can be circumscribed around a quadrilateral with angles of 68, 90, 90, and 112 degrees, as the sum of the opposite angles equals 180 degrees. The answer is True.
Step-by-step explanation:
The question is about the possibility of circumscribing a circle around a quadrilateral with the given angles of 68, 90, 90, and 112 degrees. For a circle to be circumscribed around a quadrilateral, the quadrilateral must be a cyclic quadrilateral, which means that the opposite angles must sum to 180 degrees. In this case, we can see that the sum of each pair of opposite angles is 68 + 112 = 180 degrees and 90 + 90 = 180 degrees, which satisfies the required condition. Therefore, the answer is True; a circle can be circumscribed about the quadrilateral with angles 68, 90, 90, and 112 degrees.
To determine if a circle can be circumscribed (i.e., its circumference passes through all the vertices) about a quadrilateral, we can use the following theorem:
A quadrilateral can be circumscribed about a circle if and only if its opposite angles are supplementary.
In this case, the given angles are 68, 90, 90, and 112 degrees. Let's check if the opposite angles are supplementary:
Opposite angles to 68 degrees: 90 + 90 = 180 degrees (supplementary)
Opposite angles to 112 degrees: 90 + 90 = 180 degrees (supplementary)
Since both pairs of opposite angles are supplementary, the given quadrilateral satisfies the condition for a circle to be circumscribed around it. Therefore, the statement is True.