Question

recognizing sss situations
ab ≅ bc and ad ≅ cd
what additional information would make it immediately possible to prove that triangles axb and cxb are congruent using the hl theorem?
what additional information would make it immediately possible to prove that triangles axd and cxd are congruent using the sss congruence theorem?

Explanation:

Step1: Recall HL - Hypotenuse - Leg theorem

For right - triangles, we need a right - angle and equal hypotenuses and equal legs. In \(\triangle AXB\) and \(\triangle CXB\), we already have \(AB = BC\). We need \(\angle AXB=\angle CXB = 90^{\circ}\) (right - angles) and \(BX\) is common. So the additional information for HL is \(\angle AXB=\angle CXB = 90^{\circ}\).

Step2: Recall SSS - Side - Side - Side theorem

For \(\triangle AXD\) and \(\triangle CXD\), we already have \(AD = CD\) and \(XD\) is common. We need \(AX = CX\) to satisfy the SSS congruence criterion.

Answer:

1. \(\angle AXB=\angle CXB = 90^{\circ}\)
2. \(AX = CX\)