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Advanced Geometry
Inductive Reasoning
Lesson 2
Angles
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Rays Drawn as: a line with one endpoint and one arrowhead
Named by: 2 points and a ray symbol
the endpoint must be named first
the arrow always points to the right
ND
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Opposite rays are two collinear rays that point in opposite directions.
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AngleParts of an Angle
vertex – the common endpoint
sides – the rays
B
Regionsinterior exterior
on the angle
BA and BC����������������������������
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Naming an Angle
- the number inside the angle.
Use an angle symbol and…
- three points with the vertex in the middle.
- the vertex point. *You can only use this method if there is ONE angle at the vertex.
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Examples:
Name all angles that have B as a vertex.
Name the sides of
Write another name for
5.
6., ,
,
EBD DBE
FBD OR DBF
BGand BE����������������������������
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Angle Classifications
• Acute angle – measure is less than 90°
• Right angle – measures exactly 90°
• Obtuse angle – measure is greater than 90° and less than 180°
• Straight angle – an angle with a degree measure of exactly 180° (a.k.a. straight line)
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Angle Bisector
bisects RPSPQ��������������
RPQ QPS
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Examples: In the figure, and are opposite rays, and bisects
QP��������������
QR��������������
QT��������������
.RQS
If 6 5, and 7 2,
find .
m RQT x m SQT x
m RQT
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Adjacent angles are beside each other.
They share a vertex and side.
They do not overlap.
Angle Relationships
Adjacent Angles
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Vertical angles are congruent.
Vertical Angles
Vertical angles are NOT adjacent.
They share a vertex only.
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Remember: If two rays are opposite rays, they create a
straight line.
Linear Pair
Angles of a linear pair ARE adjacent.
They share a vertex and a side.
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two angles that form a linear pair
Example: Name an angle pair that satisfies each condition.
two acute vertical angles
VZX and XZW
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two angles
sum of 90
Complementary Angles
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LINEAR PAIRS AREALWAYS
SUPPLEMENTARY.
two angles
sum of 180
Supplementary Angles
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Example:
If BCE and ECD are supplementary, find ECF and FCD.
m
m
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Find the measures of two supplementary angles if the difference in the measures of the two angles is 32.
Example:
The complement of an angle is 3.5 times smaller than the supplement of the angle. Find the measure of the angle.
106 74and
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1 90m 2 90m 3 90m 4 90m
a
b
a b
Perpendicular Lines
Symbol:
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Example: Find x so that .KO HM���������������������������������������� ���