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GIVEN: \newlineFS‾⊥HA‾\overline{FS} \perp \overline{HA}, FS‾⊥LG‾\overline{FS} \perp \overline{LG}, FLSHFLSH is a parallelogram PROVE: \newline△LGS≅△HAF\triangle LGS \cong \triangle HAF

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Q. GIVEN: \newlineFS‾⊥HA‾\overline{FS} \perp \overline{HA}, FS‾⊥LG‾\overline{FS} \perp \overline{LG}, FLSHFLSH is a parallelogram PROVE: \newline△LGS≅△HAF\triangle LGS \cong \triangle HAF
  1. Identify Given and Prove: Identify the given information and what needs to be proven.\newlineGiven: FSFS is perpendicular to HAHA, FSFS is perpendicular to LGLG, and FLSHFLSH is a parallelogram.\newlineProve: Triangle LGSLGS is similar to triangle HAFHAF.
  2. Parallelogram Properties: Use the properties of a parallelogram to determine the relationship between opposite sides and angles.\newlineIn parallelogram FLSHFLSH, opposite sides are equal and opposite angles are equal. Therefore, FL=HSFL = HS and LS=FHLS = FH. Also, angle FLSFLS is equal to angle HSFHSF.
  3. Perpendicular Lines: Use the given perpendicular lines to determine right angles.\newlineSince FSFS is perpendicular to HAHA and LGLG, angles FSHFSH and FSLFSL are right angles (9090 degrees).
  4. Additional Right Angles: Determine additional right angles based on the properties of a parallelogram.\newlineSince FLSHFLSH is a parallelogram and angle FLSFLS is equal to angle HSFHSF, angle HSFHSF is also a right angle. Therefore, angle HAFHAF is a right angle because it is vertical to angle HSFHSF.
  5. Compare Angles: Compare the angles in triangles LGSLGS and HAFHAF. Both triangles have a right angle: angle LGSLGS in triangle LGSLGS and angle HAFHAF in triangle HAFHAF. Since FLSHFLSH is a parallelogram, angle LSFLSF is equal to angle HSFHSF. Therefore, angle LGSLGS is equal to angle HAFHAF.
  6. AA Similarity Postulate: Use the Angle-Angle (AA) similarity postulate to prove the triangles are similar.\newlineSince two angles of triangle LGSLGS are equal to two angles of triangle HAFHAF (angle LGSLGS is equal to angle HAFHAF and both have a right angle), by the AA postulate, the triangles are similar.

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