Finding Angle Measures

Geometry involves scores of rules about shapes, angles, and figures.  Once you've memorized those rules, the trickiest part of geometry is figuring out how to put different rules together to solve a problem.

Some of the most common types of questions that tests use to test students geometry knowledge ask about angles.  There are many rules about angles in geometry and good geometry students learn to use lots of those rules in a single problem. 

These are some of the most common geometry rules used to determine the measure of angles in a figure:

  • Complementary angles: Angles that add up to 90°.  Angles that form right angles are always complementary.
  • Supplementary angles: Angles that add up to 180°.  
  • Linear angles: Angles that form a line are always supplementary and always add up to 180°.
  • Triangle angles: The angles of a triangle always add up to 180°.
  • Quadrilateral angles: The angles of quadrilaterals always add up to 360°.
Angle problems usually provide a figure with some measures filled in.  Students have to use the rules above to fill in the missing measures. 
Supplementary AnglesAngles in a triangle

These two angles are linear angles (together they form a line) so they add up to 180°.

$$\eqalign{120+x&=180\\-120\quad & \;\;-120\\x&=60^{\circ}}$$

The angles of a triangle add up to 180°.

$$\eqalign{90+ 50 + x &= 180\\140 + x &= 180\\-140\quad&\;\;-140\\x&=40^{\circ}}$$.

More complicated problems will ask you to apply several rules to one figure.  There is usually more than one way to solve these kinds of problems. Take them step by step.  Fill in what you know about the angle measures in the figure as you figure them out.  Work your way towards the angle the problem is asking about.

Example: In the triangle below, what is the value of $x$?

Triangle with exterior angle

 

First, look at what values you are given.

Then figure out what values you can figure out.  Where the base of the triangle extends, it forms a linear pair. 

$$\eqalign{110+a&=180\\-110\quad & \;\;-110\\a&=70^{\circ}}$$

Now you know that the bottom left corner of the triangle equals $70^{\circ}$.

Now, look at the figure again.  What can you figure out next?  Now you have two angles in the triangle with x as the third.  You know that there are 180 degrees in a triangle!

$$\eqalign{70 + 50+x&=180\\120+x&=180\\-120\quad & \;\;-120\\x&=60^{\circ}}$$

So, using two different rules, you solved for x.

Overall, angle problems just ask you to be patient and look carefully -- constantly asking yourself: what rules can I apply to gain more information about this figure?  As you gain practice, you'll learn to only do the necessary steps.  But when you first start out (or if a problem stumps you) fill in all of the information you can figure out.  The answer will probably appear!

Skill: