We are asked to write a paragraph proof to prove that triangles PRQ and TRS are congruent. Here is what we know.
Given:& R is the midpoint of QS and PT.
Since R is the midpoint, it divides PT and QS into two segments of the same length. This means that PR ≅ TR and QR ≅ SR. Let's include this fact in the diagram.
Let's list the congruent angles and sides of both triangles.
cc
PR ≅ TR & Side
∠PRQ ≅ ∠TRS & Included Angle
RQ ≅ RS & Side
We see that two sides and an included angle in triangle PRQ are congruent to two sides and an included angle in triangle TRS. By the Side-Angle-Side (SAS) Congruence Postulate, triangles PRQ and TRS are congruent.
△ PRQ ≅ △ TRS
Paragraph Proof
Let's now summarize our findings in one paragraph. This will be our paragraph proof of the conjecture.
Given: & R is the midpoint of QS and PT.
Prove: & △ PRQ ≅ △ TRS
Proof: Since R is the midpoint of QS and PT, we get that PR ≅ TR and QR ≅ SR. Additionally, ∠ PRQ and ∠ TRS are vertical angles, and so by the Vertical Angles Theorem we get ∠ PRQ ≅ ∠ TRS. By the Side-Angle-Side (SAS) Congruence Postulate we conclude that △ PRQ ≅ △ TRS.
