What are the values of w, x, y, and z given the angles in the triangle?

Understand the Problem

The question relates to geometry, specifically dealing with angles and possibly the properties of triangles. It appears to ask for the values of the variables 'w', 'x', 'y', and 'z' based on the given angles.

Answer

- $w = 120^\circ$, $x = 90^\circ$, $y = 150^\circ$, $z = 30^\circ$

Steps to Solve

Identify the angles and their relationships

From the diagram, we notice the angles are related to each other. We have an angle of $60^\circ$ and an angle of $30^\circ$. The sum of angles on a straight line is $180^\circ$.

Find angle $w$

Using the fact that the angles on a straight line sum up to $180^\circ$, we can express the relationship as:
$$ w + 60^\circ = 180^\circ $$
Thus,
$$ w = 180^\circ - 60^\circ = 120^\circ $$

Identify angle $x$

Since angle $x$ is adjacent to angle $60^\circ$ forming a triangle, we can express the sum of angles in that triangle: $$ 60^\circ + x + 30^\circ = 180^\circ $$
So,
$$ x = 180^\circ - 60^\circ - 30^\circ = 90^\circ $$

Identify angle $y$

Angle $y$ is supplementary to angle $30^\circ$: $$ y + 30^\circ = 180^\circ $$
Thus,
$$ y = 180^\circ - 30^\circ = 150^\circ $$

Find angle $z$

Using the same triangle relationship for angle $z$: $$ 60^\circ + z + 90^\circ = 180^\circ $$
So,
$$ z = 180^\circ - 60^\circ - 90^\circ = 30^\circ $$

$w = 120^\circ$
$x = 90^\circ$
$y = 150^\circ$
$z = 30^\circ$

More Information

This problem demonstrates the use of angle relationships in geometry, particularly how to find unknown angles using the properties of straight lines and triangles. The angles together sum up to specific values according to geometric principles.

Tips

Confusing straight angle relationships (angles on a line) can lead to incorrect values for angles like $w$.
Not remembering that the angles in a triangle must sum to $180^\circ$.

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