Exercise 8 Page 417

Let's begin by recalling the Angle Addition Postulate.

Angle Addition Postulate

If M is in the interior of ∠ ABC, then the measure of ∠ ABC equals the sum of the measures of ∠ ABM and ∠ MBC.

In other words, an angle formed by two adjacent angles has a measure that is the sum of the measures of these two adjacent angles. Now, we are asked to find the measures of ∠ KLN and ∠ NLM.

Because ∠ KLN and ∠ NLM are adjacent angles, the sum of their measures is equal to the measure of ∠ KLM by the Angle Addition Postulate. m∠ KLN+ m∠ NLM= m∠ KLM We are told that ∠ KLM is a straight angle so we know that its measure is 180^(∘). Let's substitute this and the given expressions into the equation. Then we can solve for x.

m∠ KLN+m∠ NLM=m∠ KLM

SubstituteExpressions

Substitute expressions

( 10x-5)+( 4x+3)= 180

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Solve for x

RemovePar

Remove parentheses

10x-5+4x+3=180

AddTerms

Add terms

14x-2=180

AddEqn

LHS+2=RHS+2

14x=182

DivEqn

.LHS /14.=.RHS /14.

x=13

Having solved the equation, we can calculate the individual angles by substituting x= 13 into the expressions for the unknown angles. m∠ KLN &: 10( 13)-5=125^(∘) m∠ NLM &: 4( 13)+3=55^(∘)