According to the question RT is either greater than RS, or the two segments have the same measure.
Let's consider the two possibilities separately.
Case 1: RT=RS
Let's summarize what we know about triangles â–³ RQT and â–³ RQS.
Claim
Justification
RT≅RS
Segments with equal measure are congruent.
QT≅QS
RQ is a median, so Q is a midpoint of TS.
Since RS is a common side of triangles â–³ RQT and â–³ RQS, the triangles have three pairs of congruent sides.
According to the Side-Side-Side (SSS) Congruence Postulate, this means that these two triangles are congruent.
△ RQT≅ △ RQS
We also know that corresponding angles of congruent triangles are congruent.
∠RQT≅ ∠RQS
Since ∠RQT and ∠RQS form a linear pair, their measures add to 180.
If ∠RQT and ∠RQS are congruent, this can only happen if both are right angles.
The consequence of this is that triangle â–³ RQT is a right triangle.
Case 2: RT>RS
Let's summarize what we know about triangles â–³ RQT and â–³ RQS.
Claim
Justification
RT> RS
By assumption.
QT≅QS
RQ is a median, so Q is a midpoint of TS.
Since RQ is a common side of triangles â–³ RQT and â–³ RQS, these triangles have two pairs of congruent sides.
This means that we can use the Converse of the Hinge Theorem.
RT> RS
⇓
m∠RQT> m∠RQS
Since ∠RQT and ∠RQS form a linear pair, their measures add to 180.
If m∠RQT> m∠RQS, this can only happen if m∠RQT>90 and m∠RQS<90.
The consequence of this is that angle ∠RQT is obtuse, so triangle △ RQT is obtuse.
Summary
Combining the two cases, we can see that if RT is greater or equal to RS, then triangle â–³ RQT is either obtuse or right.
