given: \\( \\angle abc \\) and \\( \\angle fgh \\) are right angles; \\( \\overline{ba} \\parallel…

given: \\( \\angle abc \\) and \\( \\angle fgh \\) are right angles; \\( \\overline{ba} \\parallel \\overline{gf} \\); \\( \\overline{bc} \\cong \\overline{gh} \\) prove: \\( \\triangle abc \\cong \\triangle fgh \\) step 1: we know that \\( \\angle abc \\cong \\angle fgh \\) because all right angles are congruent. step 2: we know that \\( \\angle bac \\cong \\angle gfh \\) because corresponding angles of parallel lines are congruent. step 3: we know that \\( \\overline{bc} \\cong \\overline{gh} \\) because it is given. step 4: \\( \\triangle abc \\cong \\triangle fgh \\) because of the asa congruence theorem. aas congruence theorem. third angle theorem. reflexive property.

given: \\( \\angle abc \\) and \\( \\angle fgh \\) are right angles; \\( \\overline{ba} \\parallel \\overline{gf} \\); \\( \\overline{bc} \\cong \\overline{gh} \\) prove: \\( \\triangle abc \\cong \\triangle fgh \\) step 1: we know that \\( \\angle abc \\cong \\angle fgh \\) because all right angles are congruent. step 2: we know that \\( \\angle bac \\cong \\angle gfh \\) because corresponding angles of parallel lines are congruent. step 3: we know that \\( \\overline{bc} \\cong \\overline{gh} \\) because it is given. step 4: \\( \\triangle abc \\cong \\triangle fgh \\) because of the asa congruence theorem. aas congruence theorem. third angle theorem. reflexive property.

Answer

Explanation:

Step 1: Recall Congruence Theorems

  • ASA (Angle - Side - Angle): Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
  • AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle.
  • Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
  • Reflexive Property: A figure is congruent to itself.

Step 2: Analyze Given Information

  • We have (\angle ABC\cong\angle FGH) (right angles, so angles), (\angle BAC\cong\angle GFH) (corresponding angles of parallel lines, so angles), and (\overline{BC}\cong\overline{GH}) (side).
  • The side (\overline{BC}) is not included between (\angle ABC) and (\angle BAC) in (\triangle ABC), and the side (\overline{GH}) is not included between (\angle FGH) and (\angle GFH) in (\triangle FGH).

Answer:

AAS congruence theorem.