Exercise 27 Page 312

Consider the diagram below. In order to find the values of the variables, we need to find the relationships between the angles and use the corresponding theorems and postulates.

Let's calculate each value one at a time.

Value of x

Looking at the diagram we can see that the angles that measure 96 and 2x lie on opposite sides of the parallel lines cut by a transversal. Therefore, they are consecutive interior angles.

Recall that, by the Consecutive Interior Angles Theorem, the consecutive interior angles formed by parallel lines and a transversal are supplementary. This tells us that these angles are supplementary and the sum of their measures is 180. 96+ 2x=180 We can solve this equation to find the value of x.

96+2x=180

SubEqn

LHS-96=RHS-96

2x=84

DivEqn

.LHS /2.=.RHS /2.

x=42

The value of x is 42 by the Consecutive Interior Angles Theorem.

Value of y

Observing the given graph once we again we can notice that the angles that measure 94 and 3y+44 are also consecutive interior angles.

Therefore, these angles are supplementary and the sum of their measures is 180. 94+ 3y+44=180 We can solve this equation to find the value of y.

94+3y+44=180

AddTerms

Add terms

3y+138=180

SubEqn

LHS-138=RHS-138

3y=42

DivEqn

.LHS /3.=.RHS /3.

y=14

The value of y is 14 by the Consecutive Interior Angles Theorem.