Question

given: quadrilateral ( abcdcong ) quadrilateral ( efgh ):
( overline {ad}cong overline {cd} )
prove: ( overline {ad}cong overline {gh} )

Explanation:

Step1: State the given congruence

Since quadrilateral \(ABCD\cong\) quadrilateral \(EFGH\), by the definition of congruent figures, all corresponding parts are congruent.

Step2: Identify the corresponding segment

\(\overline{AD}\) in quadrilateral \(ABCD\) corresponds to \(\overline{EH}\) in quadrilateral \(EFGH\). But wait, no, actually, when two quadrilaterals are congruent, each side of one quadrilateral is congruent to the corresponding side of the other quadrilateral. Given \(ABCD\cong EFGH\), the order of the letters matters. So \(A\) corresponds to \(E\), \(B\) to \(F\), \(C\) to \(G\), \(D\) to \(H\). So \(\overline{AD}\) corresponds to \(\overline{EH}\). But wait, no, hold on. Wait, the problem says "prove \(\overline{AD}\cong\overline{GH}\)" but if \(ABCD\cong EFGH\), then \(A\) corresponds to \(E\), \(B\) to \(F\), \(C\) to \(G\), \(D\) to \(H\). So \(\overline{AD}\) corresponds to \(\overline{EH}\). But maybe there was a typo. Wait, no, looking at the problem again: Given \(\overline{AD}\cong\overline{CD}\) (first given), and quadrilateral \(ABCD\cong\) quadrilateral \(EFGH\) (second given). Then by CPCTC (Corresponding Parts of Congruent Triangles (or here, quadrilaterals) are Congruent), since \(ABCD\cong EFGH\), \(\overline{CD}\cong\overline{GH}\) (because \(C\) corresponds to \(G\) and \(D\) corresponds to \(H\)). And since \(\overline{AD}\cong\overline{CD}\) (given), by transitive property of congruence (\(\overline{AD}\cong\overline{CD}\) and \(\overline{CD}\cong\overline{GH}\) implies \(\overline{AD}\cong\overline{GH}\)). But in the table:

1. \(\overline{AD}\cong\overline{CD}\) (Given)
2. Quadrilateral \(ABCD\cong\) quadrilateral \(EFGH\) (Given)
3. \(\overline{CD}\cong\overline{GH}\) (Corresponding parts of congruent quadrilaterals are congruent)
4. \(\overline{AD}\cong\overline{GH}\) (Transitive property of congruence: If \(a = b\) and \(b = c\), then \(a = c\))

Answer:

1. \(\overline{AD}\cong\overline{CD}\)
2. Quadrilateral \(ABCD\cong\) quadrilateral \(EFGH\)
3. \(\overline{CD}\cong\overline{GH}\)
4. \(\overline{AD}\cong\overline{GH}\)