We have proven that given RQ || ST and RQ ≈ ST, then □RUQ ≈ □TUS.
Proof:
Given:
RQ || ST
RQ ≈ ST
To prove:
□RUQ ≈ □TUS
Angle Relationships:
Since RQ || ST, we know that alternate interior angles are congruent: ∠RUQ ≅ ∠TUS.
Congruent Triangles:
We are given that RQ ≈ ST.
Since we established ∠RUQ ≅ ∠TUS, and RQ ≈ ST, we have two pairs of corresponding angles and a pair of corresponding sides congruent.
By the Side-Angle-Side (SAS) Similarity Theorem, we can conclude that ΔRUQ ≈ ΔTUS.
Proportional Sides:
Since ΔRUQ ≈ ΔTUS, we know that corresponding sides are proportional.
Therefore, RU/TU ≈ RQ/ST.
Substitute and Simplify:
We are given that RQ ≈ ST, so we can substitute it into the proportion.
This gives us RU/TU ≈ 1.
Simplifying the proportion, we get RU ≈ TU.
Congruent Areas:
We have proven that RU ≈ TU.
Since ∠RUQ ≅ ∠TUS, we have two pairs of corresponding angles and a pair of corresponding sides proportional.
By the Proportional Sides Theorem for Similar Triangles, we can conclude that □RUQ ≈ □TUS.
Therefore, we have proven that given RQ || ST and RQ ≈ ST, then □RUQ ≈ □TUS.