Question

analyze the diagram below and complete the instructions that follow.
find the value of x and the value of y.
a. x = 10, y = 130
b. x = 65, y = 10
c. x = 65, y = 130
d. x = 130, y = 10

Explanation:

Step1: Use the property of parallel lines and transversals (alternate - interior angles)

When two parallel lines are cut by a transversal, we know that \((2x - y)+60=(2x + y)-40\) (by the property of angles formed by parallel lines and a transversal).
First, simplify the equation:

$$
LATEXBLOCK0
$$

Step2: Use the property of consecutive - interior angles

Also, \((2x - y)+(2x + y)=60 + 40+ (2x - y)+(2x + y)\) (sum of angles on the same side of the transversal related to the parallel lines). Another way is to use the fact that for parallel lines, we can also consider the sum of angles.
Since \(y = 10\), substitute \(y\) into \((2x - y)+60=(2x + y)-40\) or use the fact that \((2x - y)+(2x + y)=60 + 40+(2x - y)+(2x + y)\) (a more straightforward approach is:
We know that \((2x - y)+60=(2x + y)-40\) (from step 1 gives \(y = 10\)). Now substitute \(y\) into \(2x-y + 60\) (or \(2x + y-40\)). Let's use another property.
The sum of the angles \(60+(2x - y)+(2x + y)+40 = 360\) (sum of angles around a point formed by the transversals with the parallel lines). Simplifying \(60+40+(2x - y+2x + y)=360\), we get \(100 + 4x=360\), \(4x=260\), \(x = 65\)

Answer:

B. \(x = 65,y = 10\)