Question

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

Answer 1

Answer:

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.