The LL Theorem: Understanding the Relationship Between Congruent Legs of Right Triangles

LL Theorem

The LL theorem, also known as the Long Leg Theorem or Leg-Leg Congruence Theorem, is a theorem in geometry that relates the sides of two right triangles

The LL theorem, also known as the Long Leg Theorem or Leg-Leg Congruence Theorem, is a theorem in geometry that relates the sides of two right triangles.

Statement of the LL Theorem: If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

In simpler terms, if the two legs of one right triangle are equal in length to the corresponding legs of another right triangle, then the triangles have all their sides equal and the angles in both triangles are also equal.

To prove the LL theorem, we can use the concept of congruent triangles.

Proof:

1. Consider two right triangles, Triangle ABC and Triangle DEF, with right angles at vertices B and E, respectively.
2. Assume that AB = DE (the legs of the triangles are congruent) and BC = EF (the hypotenuses are congruent).
3. We need to show that Triangle ABC is congruent to Triangle DEF.
4. By definition, triangles are congruent if their corresponding sides and angles are congruent.
5. Since AB = DE and BC = EF, we have two pairs of congruent sides in the triangles.
6. Additionally, the right angles at B and E are congruent.
7. To prove congruence, we need to show that the third pair of corresponding sides and angles are also congruent.
8. By the Pythagorean Theorem, in Triangle ABC, AC = √(AB^2 + BC^2), and in Triangle DEF, DF = √(DE^2 + EF^2).
9. Since AB = DE and BC = EF, the expressions for AC and DF are equal.
10. Therefore, AC = DF, which means the remaining pair of corresponding sides is congruent.
11. By the Side-Angle-Side (SAS) congruence criterion, we have shown that Triangle ABC is congruent to Triangle DEF.
12. Thus, the LL theorem is proved.

The LL theorem is useful in solving various geometry problems involving right triangles. It allows us to determine the congruence of right triangles based on the equality of their legs. This theorem is an essential concept in geometric proofs and is a fundamental principle in the study of congruent triangles.

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