If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
Note that this theorem is used to compare the sides of a triangle. For this reason, we will first compare the lengths of RM and RP, and the lengths of RQ and RP. Then, we will be able to determine the relationship between RM and RQ.
Relationship Between RM and RP
Let's highlight △ MPR.
We see that ∠ RPM is opposite to RM and ∠ PMR is opposite to RP. However, the diagram does not show a measure for ∠ RPM. To find this measure, we will first find m∠ MRP. Notice that ∠ SRM, ∠ MRP, and ∠ PRQ form a straight angle. The sum of their measures is 180^(∘).
60^(∘)+ m∠ MRP + 85^(∘) =180^(∘)
⇕
m∠ MRP =35^(∘)
As we can see, ∠ RQP is opposite to RP and ∠ QPR is opposite to RQ. Therefore, since ∠ RQP is greater than ∠ QPR, we know that RP is longer than RQ.
RP > RQ
Conclusion
We have shown the relationships between RM and RP, and RQ and RP.
RM > RP
RP > RQ
Finally, we can use the Transitive Property to conclude that RM is greater than RQ.
RM > RQ
