Exercise 66 Page 581

Let's begin with recalling the Incenter Theorem. The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. Since we are given that S is the incenter of triangle PLJ, PS, LS and JS are the bisectors of ∠ P, ∠ L and ∠ J respectively.

Since we are given that m∠ MPS is 28^(∘) and m∠ SLK is 24.5^(∘), we can evaluate m∠ P and m∠ L by multiplying the given angle measures by 2. m∠ P=2* 28^(∘)=56^(∘) m∠ L=2* 24.5^(∘)=49^(∘) Now, using the fact that the sum of the measures of each triangle is 180^(∘), we will evaluate m∠ J.

m∠ P+m∠ L +m∠ J=180^(∘)

SubstituteII

m∠ P= 56^(∘), m∠ L= 49^(∘)

56^(∘)+ 49^(∘)+m∠ J=180^(∘)

AddTerms

Add terms

105^(∘)+m∠ J=180^(∘)

SubEqn

LHS-105^(∘)=RHS-105^(∘)

m∠ J=75^(∘)

The measure of ∠ J is 75^(∘). Since SJ divides this angle into two congruent angles, the measure of ∠ SJP will be one half of the measure of 75^(∘). m∠ SJP=1/2*75^(∘)=37.5^(∘) The measure of ∠ SJP is 37.5^(∘).