Find the values of angles w, x, y, and z in the given geometric configuration.

Understand the Problem

The question is requesting us to find the values of the angles w, x, y, and z given in the geometric figure. We will apply the concepts of angle relationships, such as supplementary angles and alternate interior angles, to solve for these unknown values.

Answer

The angles are $w = 120^\circ$, $x = 30^\circ$, $y = 155^\circ$, and $z = 60^\circ$.

Steps to Solve

Using the properties of supplementary angles

The angles $60^\circ$ and $w$ are supplementary because they are on a straight line. This means their sum is $180^\circ$.

So, we have: $$ w + 60 = 180 $$

Solving for angle $w$

We can solve the equation for $w$: $$ w = 180 - 60 $$

Calculating angle $w$

Now, calculate the value: $$ w = 120^\circ $$

Finding angle $x$ using alternate interior angles

The angle $x$ is an alternate interior angle to the angle $30^\circ$. Therefore: $$ x = 30^\circ $$

Finding angle $y$ using supplementary angles

The angles $y$ and $25^\circ$ are also supplementary: $$ y + 25 = 180 $$

Solving for angle $y$

Now solve for $y$: $$ y = 180 - 25 $$

Calculating angle $y$

Now, calculate: $$ y = 155^\circ $$

Finding angle $z$ using alternate interior angles

The angle $z$ is an alternate interior angle to the angle $60^\circ$: $$ z = 60^\circ $$

The angles are:

$w = 120^\circ$
$x = 30^\circ$
$y = 155^\circ$
$z = 60^\circ$

More Information

In this problem, we utilized the properties of complementary and supplementary angles to find the unknown angle measures. The understanding of angle relationships is crucial in many geometric problems.

Tips

Confusing alternate interior angles: Make sure to visualize or draw the relationships correctly.
Not using the sum of angles correctly: Remember that supplementary angles always add up to $180^\circ$.

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