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LF || KG

K is the midpoint of LH
Which theorem will prove that LFK is congruent to KGH?

ASA

AAS

SAS

SSA

SSS
H&k are square angles F&G are right lines l&k are are left lines

1 answer

To determine which theorem can prove that triangle LFK is congruent to triangle KGH, we need to analyze the information given.

From your statement:

  • K is the midpoint of LH, which implies that LK = KH.
  • Given that H and K are square angles (right angles), we know that angles ∠LKH and ∠KHG are right angles.
  • Points F and G are along the lines, and these lines appear to be in the same plane relative to the triangles.

Congruence Conditions Utilized:

  1. Angle-Side-Angle (ASA): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
  2. Angle-Angle-Side (AAS): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.
  3. Side-Angle-Side (SAS): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
  4. Side-Side-Angle (SSA): This condition does not imply congruence of the triangles (the ambiguous case).
  5. Side-Side-Side (SSS): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

Conclusion:

In this case:

  • We have two right angles (∠LKH and ∠KHG).
  • We have the segment equivalency where LK = KH (because K is the midpoint).

Using the SAS theorem:

  • Side LK is equal to side KH.
  • The included angle between them (∠LKH) is a right angle, which is equal to ∠KHG.
  • Applying SAS proves that triangles LFK and KGH are congruent.

Final Answer:

The theorem that will prove that triangle LFK is congruent to triangle KGH is SAS (Side-Angle-Side).