Exercise 64 Page 627

Let's start by finding the value of x. Then we will use it to calculate the value of y.

Value of x

In order to calculate the value of x, we will need to analyze the given diagram. What is the relationship between the angles measuring (4x-5)^(∘) and (3x+11)^(∘)?

Notice that these are corresponding angles. The Corresponding Angles Postulate tells us that if parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. Therefore, the angles measuring (4x-5)^(∘) and (3x+11)^(∘) are congruent and their measures are equal. 4x-5=3x+11 We can solve the above equation to find the value of x.

Value of y

Before we find the value of y, we will first find the measures of the corresponding angles. To do so, we will substitute the value of x into either one of the given expressions with x-terms. Let's use the expression 4x-5.

We found that the measures of the corresponding angles are 59^(∘). Let's mark this value on the diagram.

To find the value of y, we need to find the relationship between the angle measuring (3y+1)^(∘) and one of the other angles. Let's analyze the diagram once more.

The angles that measure (3y+1)^(∘) and 59^(∘) are consecutive interior angles. The Consecutive Interior Angles Theorem tells us that such angles are supplementary. Therefore, the sum of their measures equals 180^(∘). (3y+1)+59=180 We can solve the above equation to find the value of y.