Final answer:
By calculating the angles of the quadrilateral from the given ratio and understanding that adjacent angles between parallel sides of a trapezium are supplementary, we have demonstrated that the given quadrilateral, with angles of 36, 72, 108, and 144 degrees, satisfies the definition of a trapezium.
Step-by-step explanation:
To show that a quadrilateral with angles in the ratio of 1:2:3:4 is a trapezium, we must first understand the properties of the angles within a quadrilateral and a trapezium.
The sum of the angles in any quadrilateral is 360 degrees. Suppose the smallest angle of the quadrilateral is 'x' degrees. Then, the angles are x, 2x, 3x, and 4x due to the given ratio. Adding these together, we have:
x + 2x + 3x + 4x = 360
Simplifying, we get 10x = 360, which gives us x = 36 degrees. Now we can determine the individual angles: 36, 72, 108, and 144 degrees.
In a trapezium (or trapezoid in North American English), there is only one pair of opposite sides that are parallel. Adjacent angles between a pair of parallel sides are supplementary (sum to 180 degrees). The angles measuring 72 and 108 degrees are supplementary, suggesting that they are between a pair of parallel sides. Therefore, the quadrilateral is indeed a trapezium.