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An
angle
is the intersection of two rays with a
common endpoint.
| Remember that an
angle is named using three letters, where the middle letter
corresponds to the vertex of the angle. The angle at
the right is <ABC or <CBA. If it
is perfectly clear which angle is being named, an angle may
be referred to by its vertex letter alone, such as <B. |
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If D lies in the interior of <ABC, then
m<ABD + m<DBC =
m<ABC.
This concept is sometimes
stated as
"the whole is equal to the sum of its parts". |
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Types of Angles
(Definitions) |
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An
acute angle is an angle
whose measure is less
than 90°
A right
angle is an angle whose
measure is
90°.
An obtuse
angle is an
angle whose measure is more
than 90°,
but
less than 1800
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A
straight angle
is an angle whose measure
is
180°.
A
reflex angle
is an angle whose measure is
more
than
180°.
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Pairs of
Angles
(Definitions) |
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The definitions above apply to angles when we look at one angle alone,
but there are also some special relationships between pairs of angles.
Adjacent Angles are 2 angles that share
a common vertex, a common side and no common interior points.
(They share a vertex and share a side, but do
not overlap.) |
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<1 and
<2 are adjacent angles.
<1 and <ABC are NOT
adjacent.
(<ABC overlaps <1) |
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Vertical Angles
are 2 angles whose sides form two
pairs of opposite rays (straight lines).
Vertical angles are
not
adjacent. They are located across
from one another in the corners of the "X" formed by the
two straight lines. They are always equal in measure.
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<1 and
<3 are vertical angles.
<2 and <4 are vertical angles.
<1 and <2 are NOT vertical. |
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THEOREM:
Vertical angles are congruent.
Complementary Angles
are 2 angles
the sum of whose measures is 90°.
Complementary angles can be placed
so that they form perpendicular lines, but do not "have to be" in this
configuration. |
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<1 and <2 are
complementary.
<XYZ and <1 are NOT
complementary.
(the rays are perpendicular) |
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THEOREM:
Complements of the same angle, or congruent angles, are congruent.
Supplementary
Angles are 2 angles the
sum of whose measures is 180°.
Supplementary angles can be
placed so that they form a straight line (a linear pair), but they do not "have to be"
in this configuration.
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<1 and
<2 are supplementary.
The line passing through points A, B, and C is a straight line.
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THEOREM:
Supplements of the same angle, or congruent angles, are congruent.
A Linear Pair
is 2 adjacent angles
whose non-common sides form opposite rays. The
angles MUST be adjacent. |
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<1 and <2 form a linear pair.
The line passing through points A, B, and C is a straight line.
<1 and
<2 are supplementary. |
THEOREM:
If two angles form a linear pair, they are supplementary.
THEOREM: If
two congruent
angles form a linear pair, they are right angles.
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