Exercise 49 Page 342

Let's label the angles.

Focusing on different parts of the figure, we can make several observations.

• Angles ∠ 1 and ∠ 2 together form a straight angle.
• The expressions for the measures of angles ∠ 3 and ∠ 5 only involve the variable y.
• Angles ∠ 4 and ∠ 5 together form a straight angle.Using these observations, we can make a plan. We will first find m∠ 2, then y, then z.

  Finding m∠ 2

  Notice that angles ∠ 1 and ∠ 2 together form a straight angle, so their measures add to 180^(∘).

  

  Since the measure of ∠ 1 is given, this lets us find the measure of m∠ 2.

  m∠ 1+m∠ 2=180

  Substitute

  m∠ 1= 135

  135+m∠ 2=180

  SubEqn

  LHS-135=RHS-135

  m∠ 2=45

  Let's put this new information on the figure.

  Finding y

  Let's focus now on the two expressions involving y. Notice that one of these is the measure of an exterior angle, the other is the measure of an interior angle of the triangle.

  

  We can use the Exterior Angle Theorem to set up and solve an equation for y.

  m∠ 2+ m∠ 3= m∠ 5

  SubstituteExpressions

  Substitute expressions

  45+(5y+5)=9y-2

  ▼

  Solve for x

  RemovePar

  Remove parentheses

  45+5y+5=9y-2

  AddTerms

  Add terms

  50+5y=9y-2

  AddEqn

  LHS+2=RHS+2

  52+5y=9y

  SubEqn

  LHS-5y=RHS-5y

  52=4y

  DivEqn

  .LHS /4.=.RHS /4.

  13=y

  RearrangeEqn

  Rearrange equation

  y=13

  Finding z

  We can use y=13 and the relationship between angles ∠ 4 and ∠ 5 to find z.

  

  Notice that angles ∠ 5 and ∠ 4 together form a straight angle, so their measures add to 180^(∘).

  m∠ 5+m∠ 4=180

  SubstituteII

  m∠ 5= 9y-2, m∠ 4= 4z+9

  9y-2+ 4z+9=180

  Substitute

  y= 13

  9( 13)-2+4z+9=180

  ▼

  Solve for z

  Multiply

  Multiply

  117-2+4z+9=180

  AddSubTerms

  Add and subtract terms

  124+4z=180

  SubEqn

  LHS-124=RHS-124

  4z=56

  DivEqn

  .LHS /4.=.RHS /4.

  z=14

  Answering the Question

  We found that y=13 and z=14.