Three figures exist such that \( A B C D \cong G H I J \) and \( B C D A \cong R S T U \). If \( G H=6 \mathrm{~cm}, \mathrm{HI}=8 \mathrm{~cm}, I J=10 \mathrm{~cm} \), and \( G J=9 \mathrm{~cm} \), what is \( T S \) ? 6 cm 8 cm 9 cm 10 cm

To solve for \( T S \), we start by analyzing the given congruences. Since the figures \( A B C D \) and \( G H I J \) are congruent, it means their corresponding sides are equal. Therefore, we can establish the following equalities: - \( AB = GH \) - \( BC = HI \) - \( CD = IJ \) - \( DA = GJ \) From the problem, we know: - \( GH = 6 \) cm - \( HI = 8 \) cm - \( IJ = 10 \) cm - \( GJ = 9 \) cm Since \( B C D A \) is also congruent to \( R S T U \), the corresponding sides must match up with those of triangle \( A B C D \). The corresponding sides for quadrilateral \( B C D A \) match those of \( G H I J \), so we have: - \( BC = RS \) - \( CD = ST \) - \( DA = TU \) - \( AB = UR \) To find \( T S \): - From the congruence \( B C D A \cong R S T U\), we know that \( T S \equiv CD \). Given \( CD = IJ = 10 \) cm, we find that \( T S = 10 \) cm. Thus, the answer is: 10 cm