Question

given that \\(\frac{df}{pr} = \frac{fe}{rq} = \frac{3}{2}\\), what additional information is needed to prove \\(\triangle def \sim \triangle pqr\\) using the sss similarity theorem? \\(\bigcirc\\) (de \cong pq) \\(\bigcirc\\) (\angle d \cong \angle p) \\(\bigcirc\\) \\(\frac{de}{ef} = \frac{3}{2}\\) \\(\bigcirc\\) \\(\frac{de}{pq} = \frac{3}{2}\\)

Explanation:

To prove \(\triangle DEF \sim \triangle PQR\) using the SSS (Side - Side - Side) similarity theorem, we need the ratios of all three corresponding sides of the triangles to be equal.

We are given that \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{3}{2}\). The SSS similarity theorem states that if the ratios of the three pairs of corresponding sides of two triangles are equal, then the triangles are similar.

Let's analyze each option:

• Option 1: \(DE\cong PQ\) means \(DE = PQ\), which is a congruence (equal length) and not a ratio equal to \(\frac{3}{2}\), so this does not help in proving similarity by SSS.
• Option 2: \(\angle D\cong\angle P\) is related to the AA (Angle - Angle) similarity criterion, not SSS, so this is not relevant.
• Option 3: \(\frac{DE}{EF}=\frac{3}{2}\) is a ratio of sides within \(\triangle DEF\), not a ratio of corresponding sides between \(\triangle DEF\) and \(\triangle PQR\), so this does not help.
• Option 4: \(\frac{DE}{PQ}=\frac{3}{2}\) gives the ratio of the third pair of corresponding sides (\(DE\) and \(PQ\)) equal to \(\frac{3}{2}\), which is the same as the ratios of the other two pairs of corresponding sides (\(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{3}{2}\)). So with this, we have \(\frac{DF}{PR}=\frac{FE}{RQ}=\frac{DE}{PQ}=\frac{3}{2}\), satisfying the SSS similarity theorem.

Answer:

D. \(\boldsymbol{\frac{DE}{PQ}=\frac{3}{2}}\)