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 $x$ and $120$ 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. Therefore, these angles are supplementary and the sum of their measures is

180 .

x + 120 = 180 ⇕ x = 60

The value of

x

is

60

by the Consecutive Interior Angles Theorem.

Value of $y$

Before we find the value of

y,

we can use that

x = 60

to have fewer unknown angle measures. Let's calculate the measure of

(3 x - 70) .

(3 x - 70)

Substitute

$x = 60$

(3 (60) - 70)

▼

Evaluate

Multiply

(180 - 70)

SubTerm

Subtract term

110

The measure of this angle is

11 0^{\circ} .

Let's mark this value on the diagram.

Now we can see that the angles that measure $3 y + 40$ and $110$ are supplementary angles.

Let's recall that, by the Supplement Theorem, the sum of the measures of the supplementary angles is

180 .

(3 y + 40) + 110 = 180

Now let's solve the equation for

y .

(3 y + 40) + 110 = 180

▼

Solve for

y

RemovePar

Remove parentheses

3 y + 40 + 110 = 180

AddTerms

Add terms

3 y + 150 = 180

SubEqn

$LHS - 150 = RHS - 150$

3 y = 30

DivEqn

$LHS / 3 = RHS / 3$

y = 10

The value of

y

is

10

by the Supplement Theorem.

Value of x

Value of y

Value of $x$

Value of $y$

Exercise