Question

writing a congruency statement
trapezoid cdef was reflected across the x - axis followed by a 90 rotation about the origin to create the other trapezoid shown on the graph. which congruency statement applies to the trapezoids?
cdef ≅ nmop
cdef ≅ npom
cdef ≅ morn
cdef ≅ mnpo

Explanation:

Step1: Understand congruence and transformation

Congruent figures have the same shape and size. Transformations like reflection and rotation preserve congruence. We need to match the vertices of the two trapezoids.

Step2: Analyze vertex - to - vertex correspondence

When trapezoid \(CDEF\) is reflected across the \(x -\)axis and then rotated \(90^{\circ}\) about the origin:

• Let's consider the order of vertices. For trapezoid \(CDEF\), after transformation, the vertex \(C\) corresponds to \(M\), \(D\) corresponds to \(O\), \(E\) corresponds to \(P\), and \(F\) corresponds to \(N\). But in terms of congruence statement order (the order of vertices in the congruence statement should match the correspondence of sides and angles), if we consider the trapezoid \(CDEF\) and the other trapezoid:
• If we assume the vertices of the original trapezoid \(CDEF\) and the transformed trapezoid. When we check the congruence statement \(CDEF\cong NMOP\):
• Reflection and rotation are rigid motions. Rigid motions (isometries) preserve the lengths of sides and measures of angles.
• For two congruent polygons \(ABCDE\) and \(A'B'C'D'E'\), \(AB = A'B'\), \(BC=B'C'\), \(\angle ABC=\angle A'B'C'\) etc.
• Let's assume the side - side - side (SSS) or side - angle - side (SAS) etc. congruence criteria (for polygons, the order of vertices in the congruence statement matters as it represents the correspondence of sides and angles).
• If we consider the trapezoid \(CDEF\) and \(NMOP\):
• After reflection across the \(x -\)axis and \(90^{\circ}\) rotation about the origin, the side \(CD\) (of trapezoid \(CDEF\)) corresponds to the side \(NO\) (of trapezoid \(NMOP\)) in terms of length (since rigid motions preserve length). Similarly, other corresponding sides and angles match when we write the congruence statement as \(CDEF\cong NMOP\).

Answer:

\(CDEF\cong NMOP\)