Sec. 1-3 – Angles
Def. of Angle
Two distinct rays form an angle if and only if they share only a
common endpoint.
In other words, an angle is the union of two rays which share only a
common endpoint.
The rays which form the angles are known as sides of the angle.
Remember, the sides of an angle are rays, not points.
The common endpoint is called the vertex.
Naming an angle
An angle can be named in three ways:
1) Use the vertex and one point from each side of the angle.
Notice the “<” symbol is used to denote an angle.
Notice the vertex is always located in the middle of the angle
name and one point from each side comes either first or last.
Example: If an angle is formed by AB and AC , then the angle
would be named <BAC or <CAB.
2) Use only the vertex.
The “<” symbol is followed by a single capital letter, the vertex.
This method can be used if only one angle in the diagram has
the indicated vertex.
3) Use a number placed just “inside” the angle near the vertex.
This method is often used
in more complicated diagrams.
The “<” symbol is followed by a number.
If the rays that form an angle are collinear, then their union actually
forms a line. In this case, the rays that form the angle are called
opposite rays.
Def. of Opposite Rays
Two distinct rays are considered opposite rays if and only if
they are collinear and they share ONLY a common endpoint.
An angle formed by opposite rays is known as a straight angle.
Dividing the Plane
Like a line, an angle divides a plane into 3 distinct sets of points:
1) the points that lie “inside” the angle,
2) the points that lie “outside” the angle, and
3) the points that lie on the angle itself.
The interior of an angle contains all points that lie “inside” the <.
The exterior of an angle contains all points that lie “outside” the <.
Angle Measure
Recall that in Section 1- 2, we used the Number Line Postulate
to pair every point on a line with exactly one real number and
the Ruler Postulate to calculate the distance between two points.
In Section 1-3, we extend these ideas to angles.
Angle Measure Postulate
For every angle, there is a unique positive number between 0 and
180 assigned to the angle.
This number is called the degree measure of the angle.
In Ch. 2, we used a ruler to find the length (measure) of a segment.
In Ch. 3, we use a protractor to find the measure of an angle.
Protractor Postulate
On a plane, given AB and a number r between 0 and 180, there is
exactly one ray on each side of AB such that the degree measure
of the angle formed is r.
The symbol “m<A” is read the as “the measure of angle A.”
The lower case letter “m” in front of an angle symbol denotes
“measure”
Using a protractor to measure an angle
1) Place the “center point” of the protractor on the vertex of the <
2) Place a “zero line” on one side (ray) of the angle.
3) On the appropriate scale, read the number where the other side
of the angle crosses the scale.
Classifying Angles
We often classify angles according to its measure with relation to
the critical values 0, 90, and 180.
An angle is a right angle if and only if its measure is exactly 90
degrees.
<A is a right angle iff m<A = 90
An angle is an acute angle if and only if its measure is greater
than 0 degrees but less that 90 degrees.
<A is an acute angle iff 0 < m<A < 90
An angle is an obtuse angle if and only if its measure is greater
than 90 degrees but less than 180 degrees.
<A is on obtuse angle iff 90 < m<A < 180
What would a 180o angle look like?
An angle is a straight angle if and only if its measure is exactly
180 degrees.
Notice that a straight angle is formed by a pair of opposite rays.
Thus a straight angle actually forms a line.
What would a 0o angle look like?
Since we can not identify two distinct rays, we would not form an
angle. Thus, we do not define a zero degree angle.