Question

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 know \(AB\cong BC\). We need \(\angle AXB=\angle CXB = 90^{\circ}\) (to make them right - triangles) so that we can use the HL theorem.

Step2: Recall SSS - Side - Side - Side theorem

For \(\triangle AXD\) and \(\triangle CXD\), we know \(AD\cong CD\). We also have the common side \(XD\). We need \(AX\cong CX\) to satisfy the SSS congruence theorem.

Answer:

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