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Unit 21
ANGULAR GEOMETRIC PRINCIPLES
NAMING ANGLES


Angles are named by a number, a letter, or three letters
For example, the angle shown can be called 1, B, ABC,
or CBA
A
B
1
C


When an angle is named with three letters, the vertex must
be the middle letter.
In cases where a point is the vertex of more than one angle,
a single letter cannot be used to name an angle
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TYPES OF ANGLES






An acute angle is an angle that is less than 90°
A right angle is an angle of 90°
An obtuse angle is an angle greater than 90° and
less than 180°
A straight angle is an angle of 180°
A reflex angle is an angle greater than 180° and less
than 360°
Two angles are adjacent if they have a common
vertex and a common side
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ANGLES FORMED BY A TRANSVERSAL



Transversal: A line that intersects (cuts)
two or more lines.
Alternate interior angles: Pairs of interior
angles on opposite sides of the transversal
and have different vertices.
Corresponding angles: Pairs of angles,
one interior and one exterior, located on
same side of the transversal, but with
different vertices.
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ANGLES CREATED BY A TRANSVERSAL
l
l is the transversal
1 2
4 3
5 6
8 7
angles 3 & 5
angles 4 & 6
are pairs of alternate interior angles
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ANGLES CREATED BY A TRANSVERSAL
l
l is the transversal
1 2
4 3
angles 1 & 5, 2 & 6, 3&7, 4&8
are pairs of corresponding angles
5 6
8 7
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THEOREMS AND COROLLARIES


A theorem is a statement in geometry that
can be proved.
A corollary is a statement based on a
theorem.

A corollary is often a special case of a theorem
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THEOREMS AND COROLLARIES (Cont)

The following are theorems and corollaries used
throughout the text. They are numbered for easier
reference (follow in your book for a visual)
1.
2.
3.
If two lines intersect, the opposite, or vertical angles are
equal.
If two parallel lines are intersected by a transversal, the
alternate interior angles are equal
If two lines are intersected by a transversal and a pair of
alternate interior angles are equal, the lines are parallel.
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THEOREMS AND COROLLARIES (Cont)
4.
5.
6.
7.
If two parallel lines are intersected by a
transversal, the corresponding angles are equal
If two lines are intersected by a transversal and
a pair of corresponding angles are equal, the
lines are parallel
Two angles are either equal or supplementary if
their corresponding sides are parallel
Two angles are either equal or supplementary if
their corresponding sides are perpendicular
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ANGULAR MEASURE EXAMPLE
• Determine the measure of all the missing angles in the figure
below given that l  m, p  q, 1 = 110°, and 2 = 80°:
q
1
5 3
l
m
4
6
2
• 3 = 110 because it is vertical to 1
(Theorem #1)
• 4 = 110 because it is alternate interior to
3 (Theorem #2) and corresponding to 1
(Theorem #4)
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p
• 5 = 70 (180° – 110°) because it is
supplementary to both 1 and 3
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ANGULAR MEASURE EXAMPLE
• Determine the measure of all the missing angles in the figure
below given that l  m, p  q, 1 = 110°, and 2 = 80°:
q
1
5 3
l
m
4
6
2
8
p
• 6 = 70 because it is corresponding to
5 (Theorem #4)
• 8 = 2 = 80 because two angles
are either equal or supplementary if
their corresponding sides are parallel
(Theorem #6)
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PRACTICE PROBLEMS
Define the terms in problems 1–6:

1.
2.
3.
4.
5.
6.
Obtuse angle
Reflex angle
Corresponding angles
Transversal
Straight angle
Name 1 in the figure below in three
C
additional ways:
D
1
E
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PRACTICE PROBLEMS (Cont)
7.
Determine the measure of angles 2–8 in the
figure below given that l  m and that 1 =
50°
2 3
1 4
56
7 8
l
m
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PROBLEM ANSWER KEY
1.
2.
3.
4.
5.
6.
7.
An angle greater than 90 and less than 180
An angle greater than 180° and less than 360°
A pair of angles, one interior and one exterior. Both
angles are on the same side of the transversal with
different vertices
A line that intersects (cuts) two or more lines
An angle of 180°
D, CDE, EDC
2 = 130, 3 = 50, 4 = 130, 5 = 130, 6 = 50,
7 = 50, and 8 = 130
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