The congruence theorem that can be used to prove AMNP - AABC is
O SSS.
O ASA
SAS.
O AAS.

Part A: ΔGBC is similar to ΔPBE

Part B:

Two triangles are similar if they satisfy the Angle-Angle (AA) similarity criteria

∠CGB ≅ ∠PBE given that they are vertically opposite angles

∠EGC ≅ ∠REG given that they are alternate interior angles of two parallel

∠EGC ≅ ∠BGC by reflexive property

∠REG ≅ ∠PEB by reflexive property

∴ ∠BGC ≅ ∠PEB by transitive property

∴ ΔGBC is similar to ΔPBE by Angle-Angle (AA) similarity criteria

Part C:

For the similar triangles ΔGBC and ΔPBE, we have;

BE/BG = PE/GC = PB/BC

∴ PE/400 = 250/350

PE = 400 × 250/350 ≈ 285.71

PE  ≈ 285.71

Similarly

BE/450 = 250/350

BE = 450 × 250/350 ≈ 321.43

BE ≈ 321.43

Step-by-step explanation: