Exercise 35 Page 431

In order to determine the relationship between RP and MP, we can use Theorem 5.10 about Angle-Side Relationships in Triangles.

Angle-Side Relationships in Triangles

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. Therefore, because RP and MP are two sides of △ MRP, we will consider the interior angles of △ MRP. Keeping this information in mind, let's consider the given diagram.

We can see that ∠ PMR is opposite to RP and ∠ MRP is opposite to MP. However, the diagram does not show a measure for ∠ MRP. To find this measure, notice that ∠ SRM, ∠ MRP, and ∠ PRQ form a straight angle. The sum of their measures is 180^(∘). m∠ SRM + m∠ MRP + m∠ PRQ =180^(∘) From the diagram, we know that ∠ SRM= 60^(∘) and ∠ PRQ= 85^(∘). Let's substitute these values into this equation and calculate the measure of ∠ MRP.

m∠ SRM + m∠ MRP + m∠ PRQ =180^(∘)

SubstituteII

m∠ SRM= 60^(∘), m∠ PRQ= 85^(∘)

60^(∘) + m∠ MRP + 85^(∘) =180^(∘)

AddTerms

Add terms

145^(∘) + m∠ MRP =180^(∘)

SubEqn

LHS-145^(∘)=RHS-145^(∘)

m∠ MRP =35 ^(∘)

We can add this piece of information to the diagram.

Because ∠ PMR is greater than ∠ MRP, we know that RP is longer than MP. RP > MP