Question

at an archeological site, the remains of two ancient step pyramids are congruent. if ( abcdcong efgh ), find ( mangle bad ). (diagrams are not to scale.)

( mangle bad=square )
(do not include the degree symbol in your answer.)

Explanation:

Step1: Use the property of congruent figures

Since \(ABCD\cong EFGH\), corresponding angles are equal. That is, \(\angle BAD\) corresponds to \(\angle FEH\).

Step2: Analyze the angle sum in a trapezoid - like structure (or use the property of congruent angles)

In the second figure, for the angle at \(E\) (which corresponds to \(\angle BAD\) in the first figure), we can note that if we consider the angle relationships. But more directly, because of the congruence \(ABCD\cong EFGH\), the measure of \(\angle BAD\) is equal to the measure of \(\angle FEH\).
Looking at the second pyramid (since \(ABCD\cong EFGH\)), we know that the sum of angles in a quadrilateral (if we consider the side - face as a quadrilateral - like shape, but more simply, using the congruence of the two step - pyramid structures). The angle \(\angle BAD\) and the angle at \(E\) (\(\angle FEH\)) are corresponding angles. And from the given information in the second figure (where we can assume the non - labeled angle corresponding to \(\angle BAD\) is calculated based on the congruence property). Since the two step - pyramids are congruent, \(\angle BAD\) has the same measure as the non - \(111^{\circ}\) and non - \(69^{\circ}\) angle in the congruent part. If we assume a trapezoid - like side (sum of adjacent angles in a trapezoid: \(180^{\circ}\) for adjacent angles along the non - parallel sides). But more straightforwardly, since \(ABCD\cong EFGH\), and by the order of the congruence \(A\) corresponds to \(E\), \(B\) to \(F\), \(C\) to \(G\), \(D\) to \(H\). So \(\angle BAD=\angle FEH\). And if we consider that in a congruent pair, \(\angle BAD = 69\) (by matching the non - \(111^{\circ}\) angle on the congruent side - face).

Answer:

69