A quadrilateral has vertices \( A(4,5), B(2,4), C(4,3) \), and \( D(6,4) \). Which statement about the quadrilateral is true A. \( A B C D \) is a parallelogram with noncongruent adjacent sides. B. \( A B C D \) is a trapezoid with only one pair of parallel sides. C. \( A B C D \) is a rectangle with noncongruent adjacent sides. D. \( A B C D \) is a square. E. \( A B C D \) is a rhombus with non-perpendicular adjacent sides.

To analyze the quadrilateral with vertices \( A(4,5) \), \( B(2,4) \), \( C(4,3) \), and \( D(6,4) \), we can calculate the slopes of each side. 1. The slope of \( AB \): \((4-2) / (5-4) = 2\) 2. The slope of \( BC \): \((4-4) / (2-4) = 0\) (horizontal line) 3. The slope of \( CD \): \((4-3) / (6-4) = 0.5\) 4. The slope of \( DA \): \((5-4) / (4-6) = -0.5\) The segments \( AB \) and \( CD \) have different slopes and are not parallel. Segments \( BC \) and \( AD \) are also not parallel as they have different slopes. Additionally, no opposite sides are equal in length nor are the ages perpendicular to each other. Thus, the correct statement is that **\( ABCD \) is a trapezoid with only one pair of parallel sides.** Here’s a little historical tidbit: trapezoids can be traced back to ancient civilizations, where they were studied in relation to land measurement and architecture. The Greeks, particularly Euclid, spent time categorizing shapes based on their properties, laying the groundwork for modern geometry! For real-world applications, understanding the properties of quadrilaterals like trapezoids is crucial in fields like architecture and engineering. For instance, when designing bridges or roofs, engineers utilize these shapes to ensure stability and aesthetic appeal, applying geometric principles to create efficient structures that withstand various forces.