Let's begin by analyzing the given information and the desired outcome of our proof. We want to show that △ PRS is congruent to △ QTS. Recall that by the definition of congruent figures, we want to show that the sides and the angles of these triangles are congruent.
Congruent sides
Congruent angles
PS ≅ SQ
∠ P ≅ ∠ Q
PR ≅ TQ
∠ PSR ≅ ∠ TSQ
RS ≅ ST
∠ R ≅ ∠ T
As we can see, we are given that PS ≅ SQ and PR ≅ TQ. Therefore, we need to prove that RS ≅ ST, and that the angles are also congruent.
Notice that we are given that the segment PQ bisects RT.
Thus, by the definition of a bisector we can conclude that the sides RS and ST are congruent.
Statement 1)& RS ≅ ST Reason 1)& Definition of a bisector
Next, we can tell that ∠ PSR and ∠ TSQ are vertical angles.
Therefore, by the Vertical Angles Theorem these angles are congruent.
Statement 2)& ∠ PSR ≅ ∠ TSQ Reason 2)& Vertical Angles Theorem
We are also given that PR ∥ TQ. Thus, if we draw the lines passing through these sides, then we will get two parallel lines a and b. If we also draw a line passing through the segment RT, then we will get a transversal t that intersects these parallel lines.
Notice that ∠ R and ∠ T are alternate interior angles. Lines a and b are parallel, thus by the Alternate Interior Angles Theorem ∠ R ≅ ∠ T.
Statement 3)& ∠ R ≅ ∠ T Reason 3)& Alternate Interior Angles & Theorem
Now, we know that ∠ PSR ≅ ∠ TSQ and ∠ R ≅ ∠ T. Two angles of the triangle △ PRS are congruent to two angles of the triangle △ QTS. Therefore, by the Third Angle Theorem we can conclude that the third angles are congruent, ∠ P ≅ ∠ Q.
Statement 4)& ∠ P ≅ ∠ Q Reason 4)& Third Angles Theorem
We have shown that all the sides and the angles in the triangles are congruent!
Congruent sides
Congruent angles
PS ≅ SQ
∠ P ≅ ∠ Q
PR ≅ TQ
∠ PSR ≅ ∠ TSQ
RS ≅ ST
∠ R ≅ ∠ T
Therefore, by the definition of congruent figures, △ PRS ≅ △ QTS.
Statement 5)& △ PRS ≅ △ QTS Reason 5)& Definition of congruent figures
