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Ch 3
Angles
3.1
Angles
Opposite Rays
 Opposite Rays –
 Rays that are part of the same line and have only their
endpoint in common
XY and XZ
 Also called straight angle
Angle
 Another case when two rays share an endpoint…
 Angle –
 A figure formed by two noncollinear rays that have a
common endpoint
 Vertex – common endpoint
 Sides – rays that make up angle
 Three ways to name:
 Three points (vertex in middle)
 Vertex only
 A number
Example
 Name the angle in four ways. Then identify its
vertex and its sides.
Naming Angles
 If more than one angle share a vertex, name the
angle with three points or a number
Example
 Name all angles having D as their vertex.
Interior/Exterior
 Angle separates a plane into three regions
 Interior of the angle
 Exterior of the angle
 Angle itself
Example
 Tell whether each point is in the interior,
exterior, or on the angle.
Assignment
 P92: 1 – 32 (omit 30)
 21 – 23: if false, tell why
 Read P95, #1 – 3
3-2
Angle Measure
Angle Measure
 Degree –
 Unit that angles are measured in
 1/360th of a circle
 Symbol: °
 mPQR reads ‘the measure of angle PQR’
 Reason why there is no degree sign, because it is a measure
and not a measurement
Postulate 3-1: Angles Measure
Postulate
Protractor
 Protractor –
 Geometric tool used to measure angles and sketch
angles of given measure
Example
 Use a protractor to measure angle KLM.
Examples
 Use a protractor to
measure ∠CDF.
 Find the measure of
∠PQR, ∠PQS, and
∠PQT.
Postulate 3-2: Protractor Postulate
 Meaning: from a given ray, there are two ways to
draw an angle with a given angle measure
Example
 Use a protractor to draw an angle having a
measure of 35.
 Use a protractor to draw an angle having a
measure of 65.
Classifying Angles
 Types of Angles:
 Right Angle – measure is 90
 Acute Angle – measure is less than 90
 Obtuse Angle – measure is more than 90
Example
 Classify each angle as acute, obtuse, or right
Example
 The measure of angle A is 100. Solve for x.
 The measure of angle B is 138. Solve for x.
Assignment
 P100: 1, 2 (use P99), 4 – 25, 31 – 37
 3.1/3.2 Wkst
Ch 3 Investigation
 Those Magical Midpoint
 P102: 1 – 2
3-3
The Angle Addition Postulate
Hands-On Geometry
 P104: Follow steps, answer questions
Postulate 3-3: Angle Addition Postulate
(AA Postulate)
Example
Example
Example
 Find m∠ABC if
m∠ABD = 70 and
m∠DBC = 43.
 If m∠EBC = 55 and
m∠EBD = 20, find
m∠2.
 Find m∠ABD if
m∠ABC = 110 and
m∠2 = 36.
Angle Bisector
Example
Example
Hands-On Geometry
 P107: follow steps, answer questions
Assignment
 P108: 1 – 24, 26 – 30
3-4
ADJACENT ANGLES AND LINEAR
PAIRS OF ANGLES
Adjacent Angles
Example
Linear Pair
Example
 Name the angle that
forms a linear pair
with ∠TCM.
 Do ∠1 and ∠TCE
form a linear pair?
Justify your
answer.
Example
 The John Hancock Center in Chicago, Illinois,
contains many types of angles. Describe the
highlighted angles.
Assignment
 P112: 1, 3 – 21, 23 – 27
 Read P115
1–3
3-5
Complementary and Supplementary
Angles
Complementary Angles
Complementary Angles
Supplementary Angles
Supplementary Angles
Example
 Name two pairs of complementary angles.
 Name a pair of adjacent angles.
Example
 Name a pair of nonadjacent supplementary
angles.
 Find the measure of an angle that is
supplementary to angle BGC.
Example
 Angles C and D are supplementary. If m∠C =
12x and m∠D = 4(x + 5), find x. Then find
m∠C and m∠D.
Postulate 3-4: Supplement Postulate
Example
 If m∠3 = 115 and ∠3 and ∠4 form a linear pair,
find m∠4.
Assignment
 P119: 1 – 37
3-6
Congruent Angles
Congruent Angles
Vertical Angles
Theorem 3-1: Vertical Angle Theorem
Example
 Find x.
Theorems
Theorems
Example
 Suppose ∠A ≅ ∠B
and m∠B = 47. Find
the measure of an
angle that is
supplementary to ∠A.
 In the figure below,
∠1 is supplementary
to ∠2, ∠3 is
supplementary to ∠2,
and m2∠ = 105.
Find m∠1 and m∠3.
Example
Theorems
Assignment
 P125: 1 (with compass), 4 – 22, 24 – 28
3-7
Perpendicular Lines
Perpendicular Lines
Looking at a Proof
Theorem 3-8
Example
Example
Hands-On Geometry
 P130
 Follow steps
 Answer Questions
Theorem 3-9
 Can there be another perpendicular to a line
through a certain point?
Assignment
 P131: 1, 3 – 27, 29 – 33
Review
 P134: 1 – 39
 P137: 1 – 20
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