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SEGMENTS, RAYS, ANGLES
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DEFINITION: A line segment is a part of a line that has finite length.
The line segment with endpoints A and
B will be denoted by
AB&bar;
or
BA&bar; .
DEFINITION: A ray is a half-line, together with its endpoint.
If a ray has endpoint A , and if
B is any other point on the half-line,
then the ray
will be denoted by
AB→ .
Notice that
AB→
is different from
BA→ .
AB→ | |
BA→ |
DEFINITION: An angle is a pair of rays that share a common endpoint.
The rays are called the sides of the angle.
The common endpoint is called the vertex of the angle.
If there is only one angle with vertex V ,
then the angle can be denoted by the simple name ∠V .
Sometimes, a slightly more complicated notation is needed for angles.
If A is a point on one side,
V is the vertex, and
B is a point on the other side,
then the angle can be denoted by ∠AVB
or ∠BVA .
∠V | |
∠AVB |
A protractor is a device that is used to measure angles.
The measure of an angle is a number that specifies how "wide" the angle is.
Angles are often measured in degrees.
Take one complete revolution (once around a circle), and divide it into 360 equal parts. Each part is one degree.
Thus, one degree is 1360
of a complete revolution.
TWO VIEWS OF A ONE-DEGREE ANGLE;
1360
of a complete revolution
The notation 45° denotes 45 degrees.
You should be very comfortable with some common degree measures:
30°,
45°,
60°,
90° .
DEFINITION: A right angle is a 90° angle.
Right angles are marked in a diagram using this little square (since a square has 90° angles):
INTRODUCTION TO GEOMETRIC CONGRUENCE
An extremely important concept for comparing geometric figures is congruence.
Two geometric figures are congruent if they have exactly the same size and shape.
Congruent figures can be superimposed one over the other, and they will match up perfectly.
The symbol that is used for congruence is ≅ .
Note the following:
- In mathematics, the word distinct is used to mean different.
- CONGRUENCE OF POINTS, LINES, and PLANES:
All points are congruent.
That is, for all points A and
B ,
A≅B .
All lines are congruent.
That is, for all distinct points A and B ,
and for all distinct points C and D ,
AB↔
≅CD
↔
(read aloud as: line A-B is congruent to line C-D).
All planes are congruent.
That is, for all distinct noncollinear points A ,
B , and C ,
and for all distinct noncollinear points D ,
E , and F ,
ABC≅DEF
(read aloud as: plane A-B-C is congruent to plane D-E-F).
- CONGRUENCE OF LINE SEGMENTS:
For distinct points A and B ,
AB¯
≅BA
¯
(read aloud as: line segment A-B is congruent to line segment B-A).
Indeed, you can decide if two line segments are congruent by comparing their lengths:
-- if two line segments have the same length, then they are congruent
-- if two line segments are congruent, then they have the same length
Recall that these previous two sentences can be abbreviated as:
Two lines segments are congruent if and only if they have the same length.
If you need to review if and only if and the concept of mathematical equivalence,
then click here.
On this exercise, you will not key in your answers.
However, you can check to see if your answer is correct.
