Question

In isosceles triangle ∆ABC, BM is the median to the base AC . Point D is on BM . Prove the following triangle congruencies: ∆AMD ≅ ∆CMD.

Answer 1

See explanation

Explanation:

In isosceles triangle ∆ABC, BM is the median to the base AC.

Since ΔABC is isosceles, then AB ≅ BC.

Since BM is the median, then AM ≅ MC.

Consider triangles ABM and CBM. In these triangles,

• AB ≅ BC;
• AM ≅ MC;
• BM ≅ BM (reflexive property).

So, ΔABM ≅ ΔCBM by SSS postulate. Congruent triangles have conguent corresponding sides and angles, so

∠AMB ≅ ∠CMB.

Consider triangles AMD and CMD. In these triangles,

• ∠AMB ≅ ∠CMB;
• AM ≅ MC;
• MD ≅ MD (reflexive property),

so ΔAMD ≅ ΔCMD by SAS postulate