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Objectives:
To identify angles formed by two lines and a
transversal
To prove and use properties of parallel lines
Transversal -> a line that intersects two
coplanar lines at two distinct points
Notice, the transversal creates 8 distinct angles.
<1 and <2 are alternate interior angles
<1 and <4 are same-side interior angles
<1 and <7 are corresponding angles
*Postulate 3.1 - Corresponding Angles Postulate
*If a transversal intersects two parallel lines, then
the corresponding angles are congruent
1
2
<1 ≈ <2
*Theorem 3.1 - Alternate Interior Angles
Theorem
*If a transversal intersects two parallel lines, then
alternate interior angles are congruent
4
3
2
1
a
b
<1 ≈ <3
*Theorem 3.2 - Same-side Interior Angles
Theorem
*If a transversal intersects two parallel lines, then
same-side interior angles are supplementary
4
3
2
1
a
b
m<1 + m<2 = 180
*Theorem 3.3 - Alternate Exterior Angles
Theorem
*If a transversal intersects two parallel lines, then
alternate exterior angles are congruent
1
2
a
b
3
<1 ≈ <3
*Theorem 3.4 - Same-side Exterior Angles Theorem
*If a transversal intersects two parallel lines, then the
same-side exterior angles are supplementary
1
2
a
b
3
m<2 + m<3 = 180
c
a
d
8
7
6
2
50°
4
5
b
1
3
Given: a parallel to b
c parallel to d
Find measures of all angles
<1
<2
<3
<4
=?
=?
=?
=?
<5
<6
<7
<8
=?
=?
=?
=?
c
d
8
a
7
6
2
50°
b
m<1
m<2
m<3
m<4
m<5
m<6
m<7
m<8
4
5
1
=
=
=
=
=
=
=
=
3
50° (corresponding angles)
130° (same-side interior angles are supplementary)
130° (corresponding angle to <2)
130° (vertical angle to <3)
50° (alternate interior angle)
50° (alternate interior angle)
130° (same-side interior angles are supplementary; <6)
50 (vertical angle to 50°)
*Homework #12
*Due Wednesday Sept 19
*Page 131 – 132
*# 1- 7 odd
*# 11 – 25 odd
*
*Objectives:
To use a transversal in proving lines parallel
*Postulate 3.2 - Converse of the Corresponding
Angles Postulate
*If two lines and a transversal form
corresponding angles that are congruent, then
the two lines are parallel.
1
P
Q
2
P
Q
*Theorem 3.5 - Converse of the Alternate Interior
Angles Theorem
*If two lines and a transversal form alternate
interior angles that are congruent, then the two
lines are parallel.
P
Q
1
4
2
~
If <1 = <2, then P parallel to Q
*Theorem 3.6 - Converse of the Same-Side Interior
Angles Theorem
*If two lines and a transversal form same-side
interior angles that are supplementary, then
the two lines are parallel.
P
Q
1
4
2
If <2 and <4 are supplementary, then P parallel to Q
* Ex: Which lines, if any must be parallel if <1 congruent <2?
* Justify the answer with a theorem or postulate.
3
C
E
D 1
4 K
2
Which lines, if any, must be parallel if <3 congruent <4?
Which lines, if any, must be parallel if <3 and <2 are supplementary?
*Theorem 3.7 - Converse of the Alternate Exterior
Angles Theorem
*If two lines and a transversal form alternate exterior
angles that are congruent, then the two lines are
parallel.
P
Q
1
3
2
If <1 congruent <2, then P parallel to Q
*Theorem 3.8 - Converse of the Same-Side Exterior
Angles Theorem
*If two lines and a transversal form same-side
exterior angles that are supplementary, then
the two lines are parallel.
P
Q
1
3
2
If <1 and <3 are supplementary, then P parallel to Q
* Example using algebra
P
Q
40°
140°
(2x + 6)°
Find the value of x for which P parallel
to Q. Explain how you came to your
answer. Use both possible ways.
*Homework # 13
*Due Thursday/Friday (Sept 20/21)
*Page 137 – 138
*#1-21 odd
*
*Objectives:
To relate parallel and perpendicular lines
*Theorem 3-9
*If two lines are parallel to the same line, then
they are parallel to each other.
Given:
a II c
c II b
Therefore
a II b
*Theorem 3-10
*In a plane, if two lines are perpendicular to the
same line, then they are parallel to each other.
t
m
n
m II n
*Theorems 3-9 and 3-10 gave conditions by which you can
conclude that lines are parallel. The next theorem will
proved a way to conclude that lines are perpendicular.
*Theorem 3-11
*In a plane, if a line is perpendicular to one of two
parallel lines, then it is also perpendicular to the
other.
n
l
m
n perpendicular to m
*Homework #14
*Due Monday (Sept 24)
*Page 143 – 144
*# 1 – 21 odd
*
*Objectives:
To classify triangles and find the measures of
their angles
To use exterior angles of triangles
*Theorem 3-12 -> Triangle Angle-Sum Theorem
* The sum of the measures of the angles of a
triangle is 180.
B
A
C
m<A + m<B + m<C = 180
G
39 21
F
65
y
x
J
Find the value of x, y, and z.
z
H
* In chapter 1, we classified angles by their measures (acute, right,
obtuse). We can also classify a triangle by its angles and sides.
Equiangular
All angles congruent
Obtuse
one obtuse angle
Acute
all angles acute
Right
one right angle
Equilateral
all sides congruent Isosceles
two sides congruent
Scalene
no sides
congruent
* Exterior angle of a polygon -> an angle formed by a side and
an extension of an adjacent side
* Remote interior angles -> the two nonadjacent interior
angles for each exterior angle
1 <– Exterior angle
3
2
Remote interior angles
*Theorem 3-13 -> Triangle Exterior Angle Theorem
*The measure of each exterior angle of a triangle
equals the sum of the measures of its two remote
interior angles.
2
1
3
m<1 = m<2 + m<3
*Homework # 15
*Due Tuesday (Sept 25)
*Page 150
*# 1-20 all
*Quiz Thursday/Friday
*Section 3.1 – 3.5
*
*Objectives:
To classify polygons
To find the sums of the measures of the
interior and exterior angles of polygons
*Polygon -> a closed plane figure with at least
three sides that are segments. The sides intersect
only at their endpoints, and no adjacent sides are
collinear.
*Naming polygons -> start at any vertex and list the
vertices consecutively in a clockwise or
counterclockwise direction.
B
A
C
D
E
Polygon
Not a polygon: not a
closed figure
Not a polygon: two
sides intersect
between endpoints
The polygon above can be named in a few ways:
ABCDE, EDCBA, CDEAB, BCDEA, etc…
The polygon includes five segments
Segment: AB, BC, CD, DE, EA
The polygon includes five angles
<A, <B, <C, <D, <E
*Classifying Polygons
*Can be classified by the number of sides
* 3 – Triangle, 4 – quadrilateral, 5 – pentagon, 6 – hexagon, 7 –
heptagon, 8 – octagon, 9 – nonagon, 10 – decagon, 12 – dodecagon
*Can be classified by shape
* Convex -> has no diagonal with points outside the polygon
* Concave -> has at least one diagonal with points outside the
polygon
Concave
Convex
* Classify the following polygons by its sides. Identify each as
convex or concave
*Theorem 3-14 -> Polygon Angle-Sum Theorem
*The sum of the measures of the, angles of
an n-gon is (n - 2)180, where n is the
number of sides of the figure.
*Ex:
*Find the sum of the measures of the angles of a
15-gon
Ex:
The sum of the measures of the angles of a
given polygon is 720. How many sides did the
polygon have?
*Theorem 3-15 -> Polygon Exterior Angle-Sum Theorem
*The sum of the measures of the exterior angles of a
polygon, one at each vertex, is 360.
3
2
4
1
5
For the above pentagon:
m<1 + m<2 + m<3 + m<4 + m<5 = 360
Equilateral polygon -> all sides are congruent
Equiangular polygon -> all angles are congruent
Regular polygon -> both equilateral and equiangular
*Homework #16
*Due Wed (Sept 26)
*Page 161
*# 1 – 25 all
*Quiz Thursday/Friday
*
*Objectives:
To graph lines given their equations
To write equations of lines
*Slope-intercept form -> y = mx + b
*m = the slope of the line
*b = the y-intercept
*The y-intercept is the y-coordinate of the
point where a line crosses the y-axis. The xintercept is the x-coordinate of the point
where a line crosses the x-axis.
*By postulate 1-1 (two points determine a line), you need
only two points to graph a line.
*The y-intercept gives one point,
the slope can be used to
plot another.
*Standard form of a linear equation -> Ax + By = C, where A,
B, and C are real numbers and A and B are not both zero.
*To graph an equation written in standard form, you can
readily find two points for the graph by finding:
* The x-intercept
* The y-intercept
Ex: Graph 6x + 3y = 12
Graph -2x + 4y = -8
* Point-slope form -> for a non-vertical line through point (𝑥1 ,
𝑦1 ) with slope m is -> y - 𝑦1 = m(x - 𝑥1 )
* Ex: Using point-slope form
* Write an equation of the line through point P(-1, 4) with slope 3
* y - 𝑦1 = m(x - 𝑥1 )
* y – 4 = 3[x – (-1)]
* y – 4 = 3(x + 1)
* Write an equation of the line with slope -1 that contains point
P(2, -4).
* Write an equation of the line through A(-2, 3) and B(1, -1).
*Homework #17
*Due Tuesday (Oct 2)
*Page 169
*#1 – 27 odd
*
*Objectives:
To relate slope and parallel lines
To relate slope and perpendicular lines
*Slopes of Parallel Lines
*If two non-vertical lines are parallel, their
slopes are equal. Likewise, if the slopes of
two distinct non-vertical lines are equal,
the lines are parallel.
*Any two vertical lines are parallel.
*Ex: Are the following lines parallel? Explain.
1
*y = -2x + 5 and 2x + 4y = 9
*y =
1
- x
2
+ 5 and 2x + 4y = 20
*Ex: Write an equation for the line parallel to y = -4x +
3 that contains point (1,-2).
* Slopes of Perpendicular Lines -> if two non-vertical lines are
perpendicular, the product of their slopes is -1. Likewise, if
the slopes of two lines have a product of -1, the lines are
perpendicular.
* Any horizontal line and vertical line are perpendicular.
* Formula for finding slope:
*
m=
𝑦2 − 𝑦1
𝑥2 −𝑥1
* In order to find if two lines are perpendicular on the coordinate
plane, find the slope of each line and multiply them together. If
the answer is -1, they are perpendicular. If the answer is
anything other than -1, they are not perpendicular.
*Homework #18
*Due Wednesday (Oct 3)
*Page 177 – 178
*#1 – 31 odd