Using Angles

Introduction to Angles

A line segment is a portion of a line with two endpoints. A ray is a portion of a line with one endpoint. Line segments are named by their endpoints and rays are named by their endpoint and another point. In each case, a segment or ray symbol is written above the points. Below, the line segment is AB and the ray is CD.

When two rays meet at their endpoints, they form an angle. Depending on the situation, an angle can be named with an angle symbol and by its vertex or by three letters. If three letters are used, the middle letter should be the vertex. The angle below could be called ∠B

 or ∠ABC or ∠CBA

. Use three letters to name an angle if using one letter would not make it clear what angle you are talking about.

Angles are measured in degrees. You can use a protractor or geometry software to measure angles. Remember that a full circle has 360 degrees.

• An angle that is exactly 90∘ (one quarter of a circle) is called a right angle. A right angle is noted with a little square at its vertex.

• An angle that is more than 90∘but less than 180∘is called an obtuse angle. 

• An angle that is less than 90∘  is called an acute angle.

• An angle that is exactly 180∘ (one half of a circle) is called a straight angle.

• Two angles are complementary if the sum of their measures is 90∘.

• Two angles are supplementary if the sum of their measures is 180∘. Two angles that together form a straight angle will always be supplementary.

• When two lines intersect, many angles are formed, as shown below.

• In the diagram above ∠AEC

   and ∠AED are adjacent angles because they are next to each other and share a ray. They are also supplementary because together they form a straight angle.

• ∠AEC and ∠DEB

   are called vertical angles. You can show that vertical angles will always have the same measure.

• 
  

Identifying Angles 

• All angles in this diagram have a vertex of E. Therefore, ∠E is ambiguous because it could refer to many different angles. Use three letters with E as the middle letter to be clear about which angle you are referring to.

• ∠ABC and ∠DBC are complementary angles with m∠ABC=40∘. What is m∠DBC?

Let's look at an example problem.

• Let m∠AEC=x∘. Show that m∠DEB must also equal x∘.

• If m∠AEC=x∘then m∠AED(180−x)∘ because ∠AEC and ∠AED form a straight angle and are therefore supplementary.

• If m∠AEC=x∘,then m∠AED(180−x)∘because ∠AEC and ∠AED form a straight angle and are therefore supplementary.

• Similarly, m∠DEB[180−(180−x)]∘(180−180+x)∘=x∘. This is how you can be confident that vertical angles will always have the same measure.