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Chapter 2.notebook
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September 10, 2015
Background information:Labeling angles and lines
• Lines can be labelled using two letters • Angles can be labelled using the ∠ symbol accompanied by
three letters. The first and third letters indicate points on the two arms. The letter in the middle is the angle. Note that the first and third letters are interchangeable
A B
C D
E
F
G
H
Chapter 2 ‐ Properties of Angles and Triangles
Another way to label an angle is by using the ∠ symbol accompaniedby the angle letter alone. However, this method can only be used when there is only one angle at the vertex point. This method could not be used in the diagram above.
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The sum of the measures of the interior angles of a triangle is 1800.
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Section 2.1 - Exploring Parallel LinesParallel lines: two lines in the same plane that, no matter how far they extend, do not intersect with each other.
Parallel lines are the same distance apart at any given point.
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Transversal a line that intersects two or more lines at distinct points
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Interior Angles any angles formed by a transversal and two parallel lines that lie inside the parallel lines.
Exterior Angles any angles formed by a transversal and two parallel lines that lie outside the parallel lines.
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Corresponding Angles one interior angle and one exterior angle that are nonadjacent and on the same side of a transversal.
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List all pairs of corresponding angles in the diagram below.
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0
18010
17020
160
30
150
40
140
50
130
60
120
70
110
80
100
90
90
41°
100
80
110
70
12060
13050
14040
15030
16020
170
10180
0 70°
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Important Points
• When a transversal intersects a pair of parallel lines, the corresponding angles are equal
• Conversely, when a transversal intersects two lines and creates equal corresponding angles, the lines are parallel
a
ec
fd
b
g h
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a bcd
e fgh
NONPARALLEL LINES
When a transversal intersects a pair of nonparallel lines the corresponding angles are NOT EQUAL
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Vertically Opposite Angles angles that are opposite each other when two lines cross. Vertically opposite angles are equal.
Supplementary angles: Angles that add up to 180o
A straight angle measures 1800
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If A and B are parallel lines, determine the unknown angles. Provide a brief statement showing your reasoning.
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A
C
E
F
G
H
B
D750
If line AB is parallel to line CD, determine the indicated angles. Provide a brief statement showing your reasoning.
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Given the following diagram, predict the measures of the angles a through g. Provide a brief statement showing your reasoning.
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4.a) Identify the following
Transversal: _____Corresponding angles:
Interior angles: ___________
Exterior angles: ___________
A
B
N
O
M
X
WR
YQZT
S
P
b) Are the corresponding angles equal? Explain
A
B
N
O
M
X
W
R
Y
QZ
TS
P
c) Identify the following
Transversal: _____Corresponding angles:
Interior angles: ___________
Exterior angles: ___________
d) Are the corresponding angles equal? Explain
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130o
G
H130o
5. In the following diagrams is AB parallel to CD? Explain how you know.
a)b)
140oH
45o
A
B
C
D
R
Q
A
B
C
D
E
F
G
corresponding angles areequal therefore the lines are parallel
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Problems
Page 72 #, 5
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Section 2.2 - Angles Formed by Parallel Lines
Alternate Interior Angles two nonadjacent interior angles on opposite sides of a transversal
Alternate Exterior Angles two exterior angles formed between two lines and a transversal, on opposite sides of the transversal
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KEY POINTS
60o
110 o
110 o
60o
60o
120o
120o 120o
70o 110o
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30o
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Example 1 pg. 76
Determine the measures of a, b, c, and d.
Corresponding angles
Vertically opposite
Interior angles on same side of transversal are Supplementary
alternate interior angles
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Example 2 pg. 77
One side of a cellphone tower will be built as shown. Use the angle measures to prove that braces CG, BF, and AE are parallel.
Correspondingangles areequalthereforeCG ll AE
Alternate interiorangles areequal thereforCG II BF
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50.
50.
130
130
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Example 3
Solve for x. Then find angle measure
2x + 8
4x + 12
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Example 4
Solve for x. Then find angle measure
3x + 102x + 15
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Practice Problems
Page 78 82 #'s 1 4, 20 and 15
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75o
36o
105oFind the angle measure of each indicated angle
Page 78 82 #'s 1 4, 20 and 15
HELP with yesterday'sHOMEWORK QUESTIONS
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Practice Problems
Page 78 82 #'s 1 4, 20 and 15
ANY HOMEWORK QUESTIONSYou would like help with
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15.
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The exterior angle of a polygon is the angle that is formed by a side of a polygon and the extension of an adjacent side.
Section 2.3 - Angle Properties in Triangles
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Nonadjacent Interior Angles the two angles of a triangle that do not have the same vertex as an exterior angle. In the diagram, angles A & B are nonadjacent interior angles to ACD
A
BC D
Exterior Angle
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KEY IDEAS: Angle Properties of Triangles
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Example 1 pg. 87
In the diagram, θ MTH is an exterior angle of ΔMAT. Determine the measures of the unknown angles in ΔMAT.
M
A T H
155o
40o
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Example 2 pg. 88
RN
20P
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Example 3
Find the measure of each lettered angle.
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TRY
Find the measure of each lettered angle.
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Example 4
Find the value of x. Then find the measure of each angle.
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Example #5
Find the value of x and then find the measure of each angle.
(2x)O (2x)O
(x)O
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75o
xo
(4x54)o
Find the value of x and then find the measure of each angle.
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Problems
Pages 90 - 92 #'s 2, 3, 11, 14, 15a
GOOD AFTERNOON
HOME WORK CHECK: PLEASE HAVE READY FOR ME TO SEE
ANY HOMEWORK QUESTIONS????
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Section 2.4 - Angle Properties in Polygons
Convex Polygon a polygon in which each interior angle measures less than 1800.
Convex nonconvex (concave)
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Sides
Triangles
Angle Sum
3 4 5 6 7 6
Convex Polygons
Formula:
The sum of the measures of the interior angles of a convex polygon is
Polygon Number of Sides Number of Triangles
Sum of Angle Measures
triangle 3
quadrilateral 4
pentagon 5
hexagon 6
heptagon 7
octagon 8
2
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http://www.gov.pe.ca/photos/original/formulasheets.pdf
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Names of Different Polygons
Number of Sides Name of Polygon3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon9 Nonagon10 Decagon11 Hendecagon12 Dodecagon
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Regular Polygons Polygons in which each side is of equal length. The measure of the interior angles are equal. Example:
For regular polygons, the following formula can be used to determine the measure of each individual angle:
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Example 1Outdoor furniture and structures like gazebos sometimes use a regular hexagon in their building plan. a) What is the sum of the interior angles in the hexagon? b) Determine the measure of each interior angle of a regular hexagon.
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TRY
Determine the sum of the interior angles and the measure of each interior angle of a regular 15sided polygon (pentadecagon).
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Example 2
The sum of the measures of the interior angles of an unknown polygon is 9000. What type of polygon is it?
HEPTAGON
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Try
The sum of the measures of the interior angles of an unknown polygon is 12600. What type of polygon is it?
NONAGON
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Each Interior angle has an exterior angle that form a straight line making 180 degrees.
Clockwise Counter clockwise
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Example 3
Determine the measure of each exterior angle of a regular convex octagon.
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Practice Problems
Pgs 99 - 102
#'s 1-3, 6, 10, 11, 17
ANY QUESTIONS
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Interior AngleExterior Angle
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85o
50o
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• Know Terminology • Completed review sheet is BONUS ON TEST
• Show all your work, have completed review sheet on your desk at beginning of tomorrows class to get BONUS
BRING CALCULATOR, NO SHARING CALCULATORS
CLASS AVERAGE >70%
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