Answer:
35
Step-by-step explanation:
The problem involves proving that ∆TSL ≅ ∆TRK, given the following information:
∠TSL ≅ ∠TRK (provided in the prompt)
TK ≅ TL (provided in the prompt)
∠LTS ≅ ∠KTK (vertical angles)
There are two ways to approach this problem:
Method 1: Using AAS (Angle-Angle-Side) Congruence
We are given that ∠TSL ≅ ∠TRK and ∠LTS ≅ ∠KTK (vertical angles).
Since the two pairs of corresponding angles are congruent, we can apply the AAS congruence postulate.
AAS states that if two triangles have two pairs of corresponding angles congruent, then the triangles are congruent.
Therefore, ∆TSL ≅ ∆TRK by AAS congruence.
Method 2: Using the HL Congruence Theorem
We are given that TK ≅ TL.
We also have ∠LTS ≅ ∠KTK (vertical angles), which implies that ∠TLS ≅ ∠TRK (corresponding angles).
HL congruence states that if two triangles have a pair of corresponding sides congruent and a pair of corresponding angles acute (less than 90 degrees), then the triangles are congruent.
In this case, TK ≅ TL (corresponding sides) and ∠TLS ≅ ∠TRK (acute angles, as both are right angles).
Therefore, ∆TSL ≅ ∆TRK by HL congruence.
Both methods lead to the same conclusion that ∆TSL ≅ ∆TRK.
