Question

Sketch and prove the following for the two congruent triangles ∆ABC and ∆DEF.

The medians drawn from vertex A and D are congruent.

Label medians MA and ND ; then △ABM≅△

by reason

Answer 1

See explanation

Explanation:

Given: ΔABC ≅ ΔDEF

AM, DN - medians

Prove: AM ≅ DN

Proof:

1. Congruent triangles ABC and DEF have congruent corresponding parts:

• AB ≅ DE;
• BC ≅ EF;
• ∠ABC ≅ ∠DEF.

2. BM ≅ MC - definition of the median AM;

3. EN ≅ NF - definition of the median DN;

4. AB ≅ 2BM, EF ≅ 2EN

BM ≅ 1/2 AB,

EN ≅ 1/2 EF,

thus, BM ≅ EN

5. Consider two triangles ABM and DEN. In these triangles \:

• AB ≅ DE (see 1));
• BM ≅ EN (see 4));
• ∠ABM ≅ ∠DEN (see 1).

So, ΔABM ≅ ΔDEN by SAS postulate.

6. Congruent triangles ABM and DEN have congruent corresponding sides BM and DN.