Question

1. which set of conditions would make (\triangle abccong\triangle def) under the sss congruence postulate?(ac = df), (bc = ef), and (ca = fd)(ab = de), (bc = ef), and (mangle c=mangle f)(ab = de), (bc = ef), and (ca = fd)(ab = de), (mangle b=mangle e), and (ca = fd)

Explanation:

Step1: Recall SSS Congruence

SSS (Side - Side - Side) congruence postulate states that if three sides of one triangle are equal to three corresponding sides of another triangle, then the two triangles are congruent. For \(\triangle ABC\cong\triangle DEF\) by SSS, we need \(AB = DE\), \(BC=EF\) and \(CA = FD\) (since the order of the vertices matters, \(A\) corresponds to \(D\), \(B\) corresponds to \(E\) and \(C\) corresponds to \(F\)).

Step2: Analyze Option 1

The first option has \(AC = DF\), \(BC = EF\) and \(CA=FD\). Here, \(AC\) and \(CA\) are the same side (just written in reverse order), and this option does not match the correct correspondence of sides for \(\triangle ABC\) and \(\triangle DEF\) (it does not have \(AB = DE\)).

Step3: Analyze Option 2

The second option has \(AB = DE\), \(BC = EF\) and \(m\angle C=m\angle F\). This includes an angle, so it is not SSS (SSS requires only sides, no angles).

Step4: Analyze Option 3

The third option has \(AB = DE\), \(BC = EF\) and \(CA = FD\). This matches the SSS congruence postulate as all three corresponding sides of \(\triangle ABC\) and \(\triangle DEF\) are equal.

Step5: Analyze Option 4

The fourth option has \(AB = DE\), \(m\angle B=m\angle E\) and \(CA = FD\). This includes an angle, so it is not SSS.

Answer:

The correct option is the third one: \(AB = DE\), \(BC = EF\), and \(CA = FD\) (the option with the circle next to \(AB = DE\), \(BC = EF\), and \(CA = FD\))