Question

Ocean waves move in parallel lines toward the shore. The figure shows the path that a windsurfer takes across several waves. For this exercise, think of the windsurfer's wake as a line. If m∠1 = (5x + 3y)° and m∠2 = (5x + y)°, find x and y.

Answer 1

If you look at the figure, the angle marked red is equal to 70° because vertical angles are equal. So, that also means that the opposite angle 2 is also 70°. So, the first equation is:

70 = 5x + y

Next, the green angle as marked is equal to
Green angle = 180 - 70 = 110
Being vertical angles, angle 1 is then equal to 110°. So,the second equation is

110 = 5x + 3y

Subtract the two equations:

    5x + y = 70
 -  5x + 3y = 110
________________
           -3y = -40
              y = -40/-3 = 13.33°

Substituting y to either one of the equations,
5x + 13.33 = 70
Solving for x,
x = 11.33°

Answer 2

Angles in transversal of parallel lines are either corresponding, vertical, alternate or interior angles.

The values of x and y are 10 and 20, respectively.

The angles are given as:

$\mathbf{\angle 1 = (5x + 3y)^o}$

$\mathbf{\angle 2 = (5x + y)^o}$

From the image of the wave (see attachment), we have:

$\mathbf{\angle 3 = 70^o}$

Where:

$\mathbf{\angle 1 = 180 - \angle 3 }$  ----- corresponding angles

$\mathbf{\angle 2 =\angle 3 }$ ----- corresponding angles

So, we have:

$\mathbf{5x + 3y =180 - 70}$

$\mathbf{5x + y =70}$

Subtract the second equation from the first to eliminate x

$\mathbf{5x - 5x + 3y - y = 180 - 70 - 70}$

$\mathbf{2y = 40}$

Divide both sides by 2

$\mathbf{y = 20}$

Substitute 20 for y in $\mathbf{5x + y =70}$

$\mathbf{5x + 20 = 70}$

Collect like terms

$\mathbf{5x = 70 -20}$

$\mathbf{5x = 50}$

Divide both sides by 5

$\mathbf{x = 10}$

Hence, the values of x and y are 10 and 20, respectively.

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