Given: parallelogram HIJK
       ᐃHOI ≅ ᐃJOI 
Prove: HIJK is a rhombus

 

1. HI ≅ JI      Corresponding parts of congruent triangles HOI and JOI

2. HI ≅ JK      Opposite sides of parallelogram HIJK

3. JI ≅ JK      Things congruent to the same thing are congruent to each
                other.  (Both congruent to HI by 1 and 2)       

4. JI ≅ HK      Opposite sides of parallelogram HIJK        

5. HI ≅ HK      Things congruent to the congruent thing are congruent to each
                other.  (HI ≅ JK ny 2, HK ≅ JI by 4, and JK ≅ JI by 3) 

6. JK ≅ HK      Things congruent to the same thing are congruent to each
                other.  (Both congruent to JI by 3 and 4)

7. HIJK is a rhombus.   All 6 pairs of sides congruent.      

Edwin