Question

in the figure shown, what additional information is needed to show that △abc ≅ △def by sss? a. ( mangle c ) b. ( overline{ac} cong overline{df} ) c. ( overline{ab} cong overline{de} ) d. ( mangle e )

Explanation:

To show \(\triangle ABC \cong \triangle DEF\) by SSS (Side - Side - Side) congruence criterion, we need all three corresponding sides of the triangles to be congruent.

From the diagram:

• We can see that one pair of sides (let's say \(BC\) and \(EF\)) are marked as congruent (the single tick mark), and another pair of sides ( \(AB\) and \(DE\) are marked with double tick marks? Wait, no, looking at the diagram, in \(\triangle ABC\), \(AB\) and \(AC\)? Wait, no, in \(\triangle ABC\), side \(AB\) and \(AC\)? Wait, no, the markings: in \(\triangle ABC\), \(AB\) has two tick marks, \(BC\) has one tick mark. In \(\triangle DEF\), \(DE\) has two tick marks, \(EF\) has one tick mark. So we already have \(AB\cong DE\) (from the two - tick marks) and \(BC\cong EF\) (from the one - tick marks). The third pair of sides should be \(AC\cong DF\) to satisfy SSS.

Let's analyze each option:

• Option A: \(m\angle C\) is an angle, SSS does not involve angles, so this is not relevant.
• Option B: \(AC\cong DF\) is the third pair of sides needed for SSS. If we have \(AB\cong DE\), \(BC\cong EF\) (from the markings) and \(AC\cong DF\), then by SSS, \(\triangle ABC\cong\triangle DEF\).
• Option C: \(AB\cong DE\) is already indicated by the markings (the double tick marks), so we don't need this as additional information.
• Option D: \(m\angle E\) is an angle, SSS does not use angle measures, so this is not relevant.

Answer:

B. \(\overline{AC}\cong\overline{DF}\)