Which triangles must be congruent? given this information about δabc, δdef, and δghi:
ab = df. ab = gi. bc = hi. de = hi. m∠b = m∠d = m∠i.

Only one pair of triangle is congruent.i., ΔABC ≅ ΔGHI

Step-by-step explanation:

Given: ΔABC , ΔDEF , ΔGHI

           AB = DF , AB = GI , BC = HI , DE = HI , ∠B = ∠D = ∠I

To find: Two Congruent traingles.

We check all cases of possible congreuent triangle with given information.

1). In  ΔABC and ΔDEF

        AB = DF  ( given )

        ∠B = ∠D  ( given )

⇒ ΔABC and ΔDEF are not congruent as sufficient information is not given i.e., either a pair of side (BC=DE) or a pair of angle( ∠A=∠F).

2). In  ΔDEF and ΔGHI

        DE = HI  ( given )

        ∠D = ∠I  ( given )

⇒ ΔDEF and ΔGHI are not congruent as sufficient information is not given.i.e., either a pair of side(DF=IG) or a pair of angle(∠E=∠H).  

3). In  ΔABC and ΔGHI

        AB = GI ( given )

        BC = HI  ( given )

        ∠B = ∠I  ( given )

⇒ ΔABC ≅ ΔGHI by SAS ( Side-Angle-Side) congruence rule

Therefore, Only one pair of triangle is congruent.i., ΔABC ≅ ΔGHI