Section2

ANGLES AND THEIR PROPERTIES.

In order to finish our list of the simplest geometric definitions we shall introduce a notion of an angle. An angle is a geometric figure that consists of two rays that share a common endpoint. The common endpoint of the two rays is called the vertex of the angle. The two rays are called the sides of the angle. Usually, angles are denoted by three points that belong to an angle but sometimes they can be denoted by Greek letters

The middle letter in the angle's name is always the vertex. The other two letters are any points that belong to each side of the angle. Sometimes the word 'angle' is replaced by a mathematical notation . So, we can say either angle BAC or

The important property of an angle is the fact that each angle has a size or a degree measure. A degree measure of an angle BAC is the least amount of rotation around the vertex A that we need to fit one side over the other. The rotation is measured in degrees therefore the degree measure of an angle is measured in degrees as well. A full circle of rotation consists of 360 degrees, a half of the circle consists of 180 degrees etc. So, according to our definition of an angle, the angle BAC (as well as any other angle) can have any measure between 0 and 180 degrees inclusive. We are going to denote the measure of the angle BAC by

An angle ABC is called acute if
An angle ABC is called right if
An angle ABC is called obtuse if

Two angles ABC and DEF are called equal if they have equal degree measures:
Two angles ABC and DEF are called complementary if
Two angles ABC and DEF are called supplementary if
Finally, two angles are called vertical if they have a common vertex and sides of one angle are extensions of sides of the other.

Now that we have all necessary definitions in hands, we are ready to discuss a few important properties of angles. The first property that we are going to formulate will be accepted without a proof since on one hand the property is obvious and on the other hand no one knows how to prove it (such statements are called axioms).

Axiom 1. Given a ray and a half-plane we can always construct an angle with degree measure less than 180 degrees. The constructed angle will be unique in a sense that if we construct one more angle with the same degree measure and in the same half-plane then this second angle will coincide with the first one.

The second property that we are going to discuss will be dealing with specifics of vertical angles. This property will be proved using the definitions given above (such statements are usually called theorems).

Theorem 2.1 Vertical angles are equal (i.e. they have equal degree measures).

Proof: First of all we need to construct a picture and rewrite the statement that we need to prove using mathematical notations.

Given: a pair of vertical angles AOB and COD and a pair of vertical angles BOD and AOC.

Need to prove: or

Please note that it is enough to prove the equality for only one pair of angles (the second equality will follow automatically from the first one).

Angles AOB and BOD are supplementary. Angles COD and DOB are supplementary as well, therefore

Solving this elementary system we immediately get the desired result:

EXERCISE SET # 2

Problem 1. Find the number of obtuse angles on the picture below.

Problem 2. Please refer to the picture from Problem 1. How many pairs of vertical angles are there?