Using Circles to Find Angle Measures

Another common theme in geometry problems is circles.  Questions will often ask you to figure out the degrees of angles that are embedded in circles.  The main rule you want to remember is: Whenever a set of angles forms a circle, they add up to 360° (every circle is 360° around).

Other important rules for Circle problems:

• Circles: Contain 360°.  Whenever a set of angles make a circle, they add up to 360°. 
• Semi-Circles: Contain 180° (it's often faster to work with a half circle than a whole circle!)
• Vertical Angles: Angles that are across from each other in an X. Vertical angles are always equal to each other.
• Linear Angles: Angles that make a line add up to 180°.

Figures that make circles do not always look like circles.  Look at the figure below:

The "circle" part of this figure is missing.  But you can easily trace it in with your finger!  All of these lines radiate out from a central point, thus all of the angles formed add up to $360^{\circ}$.  You can use the fact that they all add up to 360 to solve for x.

$$\eqalign{50+100+30+50+x&=360\\230+x&=360\\-230\quad&\;\; -230\\x&=130^{\circ}}$$

Remember, if you can split a circle in half, you can also use semi-circles, which add up to $180^{\circ}$ to solve for angles (same answer, shorter problem):

$$\eqalign{50+x&=180\\-50\quad&\;\; -50\\x&=130^{\circ}}$$

As with other angle problems, you often have to combine several angle rules to solve for a missing variable.  And, as with other problems, there are often several ways to solve a problem. Just be methodical, write in values as you find them, and you'll solve your problem. 

Example: In the figure below, find the value of $x$.

There are two ways to solve this problem.  The easiest way uses the semi circle.  As you can see, with x, you have a full semi-circle.  So, you can use the following equation

$$\eqalign{80+55+x&=180\\135+x&=180\\-135\quad&\;\; -135\\x&=45^{\circ}}$$

But, what if you don't see the semi-circle?  You can also fill in the circle using the vertical angles rule.

The vertical angles rule says that angles opposite each other in an X are equal. 

Fill in the angle opposite the $80^{\circ}$ angle.

Now you have a full circle:

Now you can use the equation:

$$\eqalign{80+55+x+80+55+x&=360\\270+2x&=360\\-270\quad&\;\; -270\\2x&=90\\\div2&\;\;\div2\\x&=45^{\circ}}$$

Overall, remember that angles that radiate out from a center form a circle, which equals $360^{\circ}$.  If you can break that circle in half, a semi-circle has $180^{\circ}$.  And, vertical angles are your friends.  Fill in those values whenever you see them.  Just be patient and careful with these problems and you'll get your answer one way or the other.