Triangles ΔABC ≅ ΔBAD so that C and D lie in the opposite semi-planes of segment AB. Prove that segment CD bisects segment AD.

See explanation

Step-by-step explanation:

Triangles ΔABC and ΔBAD are congruent. So,

AB ≅ BA; AC ≅ BD; BC ≅ AD; ∠ABC ≅ ∠BAD; ∠BCA ≅ ∠ADB; ∠CAB ≅ ∠DBA.

Consider triangles AEC and BED. In these triangles,

AC ≅ BD; ∠EAC ≅ ∠EBD (because ∠CBA ≅ ∠BAD); ∠AEC ≅ ∠BED (as vertical angles).

So, ΔAEC ≅ ΔBED. Thus,

AE ≅ EB.

This means that segment CD bisects segment AD.