Exercise 37 Page 39

a We are told that

∠ R Q S

and

∠ T Q S

form a linear pair. We are also told that

m ∠ R Q S = 2 x + 4,

and

m ∠ T Q S = 6 x + 20 .

We are asked to solve for

x .

To do so, we must know that the angles of a linear pair form a straight angle. For example, in the following diagram,

∠ 1

and

∠ 2

form a linear pair.

Therefore, if

∠ R Q S

and

∠ T Q S

are a linear pair, we have that they are supplementary and their measures sum to 180.

m ∠ R Q S + m ∠ T Q S = 180

To solve for

x,

we will substitute

2 x + 4

for

m ∠ R Q S,

and

6 x + 20

for

m ∠ T Q S .

m ∠ R Q S + m ∠ T Q S = 180

SubstituteII

$m ∠ R Q S = 2 x + 4$ , $m ∠ T Q S = 6 x + 20$

2 x + 4 + 6 x + 20 = 180

AddTerms

Add terms

8 x + 24 = 180

SubEqn

$LHS - 24 = RHS - 24$

8 x = 156

DivEqn

$LHS / 8 = RHS / 8$

x = \frac{1 5 6}{8}

FracToDiv

$\frac{a}{b} = a \div b$

x = 19.5

We have found that

x = 19.5 .

b We are asked to find

m ∠ R Q S

and

m ∠ T Q S .

To do so, we will use the fact that

m ∠ R Q S = 2 x + 4, m ∠ T Q S = 6 x + 20,

and

x = 19.5,

which we found in Part A. We will find the angle measures my substituting the

19.5

in the expressions.

Angle	Expression	$x = 19.5$	Measure
$∠ R Q S$	$2 x + 4$	$2 (19.5) + 4$	$m ∠ R Q S = 43$
$∠ T Q S$	$6 x + 20$	$6 (19.5) + 20$	$m ∠ T Q S = 137$

c We are asked to show how we can check our answer. To do so, we must know that angles of a linear pair form a straight angle. Therefore, the measures of the angles have a sum of

180 .

Let's see if that is correct for the measures we have found.

m ∠ R Q S + m ∠ T Q S = ? 180

SubstituteII

$m ∠ R Q S = 43$ , $m ∠ T Q S = 137$

43 + 137 = ? 180

AddTerms

Add terms

180 = 180

We have shown that

m ∠ R Q S + m ∠ T Q S = 180

and thus our answer is correct.

Subchapter links

Lesson Check p.37

Practice and Problem-Solving Exercises p.38-40

Standardized Test Prep p.40

Mixed Review p.40