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:

For proving $\triangle AXB\cong\triangle CXB$ using HL:

• HL (Hypotenuse - Leg) Theorem:
• In right - angled triangles, if the hypotenuse and one leg are congruent, the triangles are congruent.
• We know \(AB\cong BC\) (hypotenuse).
• We need \(\angle AXB=\angle CXB = 90^{\circ}\) (to make the triangles right - angled).

For proving $\triangle AXD\cong\triangle CXD$ using SSS:

• SSS (Side - Side - Side) Congruence Theorem:
• If \(AD\cong CD\) (given), \(AX\cong CX\) (side), and \(DX = DX\) (common side).
• We already know \(AD\cong CD\). We need \(AX\cong CX\) to satisfy the SSS condition.

Answer:

• For \(\triangle AXB\) and \(\triangle CXB\) using HL: \(\angle AXB=\angle CXB = 90^{\circ}\)
• For \(\triangle AXD\) and \(\triangle CXD\) using SSS: \(AX\cong CX\)