Prove: EF = GH and EH = FG

Part (1) To prove EF = GH and EH = FG, we will use triangle congruence. Step 1: Identify congruent triangles. From the given markings, the diagonals EG and FH bisect each other at their intersection point, let's call it M. This means EM = MG and FM = MH. Consider EMF and GMH. Step 2: Prove EMF GMH. • EM = MG (Given, diagonals bisect each other) • FM = MH (Given, diagonals bisect each other) • EMF = GMH (Vertically opposite angles) Therefore, EMF GMH by the Side-Angle-Side (SAS) congruence criterion. Step 3: Conclude side equality. Since EMF GMH, their corresponding sides are equal. Thus, EF = GH. Step 4: Prove EMH FMG. Consider EMH and FMG. • EM = MG (Given) • MH = FM (Given) • EMH = FMG (Vertically opposite angles) Therefore, EMH FMG by the Side-Angle-Side (SAS) congruence criterion. Step 5: Conclude side equality. Since EMH FMG, their corresponding sides are equal. Thus, EH = FG. We have proven that EF = GH and EH = FG. Part (2) To prove ET = TG and HT = TF (assuming TI is a typo for TF), we will use triangle congruence. Step 1: Identify congruent triangles. From the given markings, EF HG and EH FG. This means EFGH is a parallelogram. The diagonals EG and FH intersect at T. Consider EFT and HGT. Step 2: List conditions for congruence. • EF = HG (Opposite sides of a parallelogram are equal) • FET = HGT (Alternate interior angles, since EF HG and EG is a transversal) • EFT = GHT (Alternate interior angles, since EF HG and FH is a transversal) Step 3: Prove EFT HGT. Therefore, EFT HGT by the Angle-Side-Angle (ASA) congruence criterion. Step 4: Conclude segment equality. Since EFT HGT, their corresponding parts are equal. Thus, ET = TG and FT = HT. Assuming the question meant HT = TF instead of HT = TI, we have proven that ET = TG and HT = TF. Send me the next one 📸