Advanced Geometry

Inductive Reasoning

Lesson 2

Angles

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.

AngleParts of an Angle

vertex – the common endpoint

sides – the rays

B

Regionsinterior exterior

on the angle

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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.

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

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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)

Angle Bisector

bisects RPSPQ��������������

RPQ QPS

Examples: In the figure, and are opposite rays, and bisects

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.RQS

If 6 5, and 7 2,

find .

m RQT x m SQT x

m RQT

Adjacent angles are beside each other.

They share a vertex and side.

They do not overlap.

Angle Relationships

Adjacent Angles

Vertical angles are congruent.

Vertical Angles

Vertical angles are NOT adjacent.

They share a vertex only.

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.

two angles that form a linear pair

Example: Name an angle pair that satisfies each condition.

two acute vertical angles

VZX and XZW

two angles

sum of 90

Complementary Angles

LINEAR PAIRS AREALWAYS

SUPPLEMENTARY.

two angles

sum of 180

Supplementary Angles

Example:

If BCE and ECD are supplementary, find ECF and FCD.

m

m

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

1 90m 2 90m 3 90m 4 90m

a

b

a b

Perpendicular Lines

Symbol:

Example: Find x so that .KO HM���������������������������������������� ���