Many geometric techniques and solutions hinge on finding the measures of a shape's angles.

It'll be helpful to think of an angle as the measure of a turning motion or rotation. When we use the unit of **degrees** to measure angles, one full turn is defined to measure exactly $360$ degrees, which is also written as $360^\circ.$

**What is the measure of a half turn?**

If you've already taken a geometry class somewhere, you might have been asked to memorize that these little angles created by extending the edges of any polygon will always add up to exactly $360$ degrees.

But, if you think about angles as a "turning" motion, you can really see and understand ** why** that's true.

Using this observation, what is the measure of the the pink-shaded angle in the figure?

If this fan of angles adds up to a half turn, what is the value of $x?$

Which value is the largest?

What can we say about $x$ and $y,$ the measures of a **vertical pair** of angles?

**right angle**, and it has a measure of $90^\circ.$ In diagrams, 90 degree angles are sometimes identified with a small square block instead of a circle-wedge.

What is the value of $x?$

Now let's combine using everything that's been introduced so far.

What is the measure of the unknown angle?