Question

Find the measure of each labeled angle as well as x and y.

Answer 1

Considering the figure:

The horizontal lines are parallel lines crossed by a transversal line, the angles with measure 3xº and 60º are corresponding angles, which means that they are congruent, then:

`3xº=60º`

From this expression, you can determine the value of x, simply divide both sides by 3:

`\begin{gathered} (3xº)/(3)=(60º)/(3) \\ x=20º \end{gathered}`

The value of x is 20º

Next, to determine the value of y, you have to work using the quadrilateral on the bottom:

Before you can determine the value of y, you have to determine the measure of the fourth angle of the quadrilateral, which I named "z" for explanation purposes.

The angle with measure 60º and z are supplementary angles, this means that their measures add up to 180º

`60º+z=180º`

From this, you can determine the value of z:

`\begin{gathered} z=180º-60º \\ z=120º \end{gathered}`

You know that the sum of the inner angles of a quadrilateral is equal to 360º, for the quadrilateral marked with red, you can express this as follows:

`360º=120º+60º+135º+(5y-5)º`

Now we can determine the value of y:

-Simplify all like terms together:

`\begin{gathered} 360º=120º+60º+135º-5º+5y \\ 360º=310º+5y \end{gathered}`

-Subtract 310º to both sides of the equal sign:

`\begin{gathered} 360º-310º=310º-310º+5y \\ 50º=5y \end{gathered}`

-Divide both sides by 5

`\begin{gathered} (50º)/(5)=(5y)/(5) \\ 10º=y \end{gathered}`

The value of y is 10º

You can determine the measure of the angle as follows:

`\begin{gathered} (5y-5)º \\ (5\cdot10-5)º \\ (50-5)º \\ 45º \end{gathered}`