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: Identify HL requirements

HL needs right triangles, hypotenuse ≅, leg ≅.

Step2: Apply to AXB and CXB

Hypotenuse AB≅BC (given). Leg BX common (≅). Need right ∠ at X: AC⊥BX.

Step3: Identify SSS requirements

SSS needs 3 sides ≅.

Step4: Apply to AXD and CXD

AD≅CD (given), XD common (≅). Need AX≅CX.

Answer:

To prove △AXB≅△CXB by HL: AC is perpendicular to BX (∠AXB and ∠CXB are right angles).
To prove △AXD≅△CXD by SSS: AX is congruent to CX.