Download 3 notes - Blackboard

Parallel Lines and Planes
Chapter 3:
3-1:
Definitions
(page 72)
(page 73)
LINES
1. Intersecting Lines - intersection occurs at
point.
2. PARALLEL LINES (||-lines): coplanar lines that do
3. SKEW LINES: lines that are
intersect.
coplanar.
PLANES
1. Intersecting Planes - intersection is a
.
2. PARALLEL PLANES (||-planes): planes that do
intersect.
LINE and PLANE
1. Line Contained in a Plane - every point on line is in the plane.
2. Line Parallel to a Plane - the line and plane do not
3. Line Intersects the Plane - the intersection occurs at one (1)
.
.
THEOREM 3-1
If two parallel planes are cut by a third plane,
then the lines of intersection are parallel.
Given: plane X || plane Y
plane Z intersects X in line l
plane Z intersects Y in line n
Prove: l || n
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
1
m
2
4
3
11 12
9 10
15 16
5
6
z
13
14
n
8
7
t
x
y
Figure A
TRANSVERSAL: a line that intersects 2 or more
Figure B
lines in different points.
examples:
Exterior Angles - “see above diagram”
examples:
Interior Angles - “see above diagram”
examples:
ALTERNATE INTERIOR ANGLES: 2 nonadjacent interior angles on
sides of the transversal.
examples:
SAME-SIDE INTERIOR ANGLES: 2 interior angles on the same side of a
.
examples:
CORRESPONDING ANGLES: 2 angles in
positions relative to 2 lines.
examples:
Assignment: Written Exercises, pages 76 & 77: 1-17 odd #’s and 23-41 odd #’s
3-2:
Properties of Parallel Lines
(page 78)
Refer to the results from exercise #18 on page 76 and the top of page 78.
POSTULATE 10
If two parallel lines are cut by a transversal, then
angles are congruent.
x
1 2
3
4
y
5
6
8 7
THEOREM 3-2
If two parallel lines are cut by a transversal,then
interior angles are congruent.
Given: k || n
transversal t cuts k and n
t
Prove: ∠1 ≅ ∠2
k
3
1
2
n
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
THEOREM 3-3
If two parallel lines are cut by a transversal,
then same-side interior angles are supplementary.
t
Given: k || n transversal t cuts k and n
k
Prove: ∠1 is supplementary to ∠4
n
1
4
2
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
THEOREM 3-4
If a transversal is perpendicular to one of two parallel lines,
then it is perpendicular to the other one also.
Given: transversal t cuts l and n
t ⊥ l ; l || n
l
1
n
2
Prove: t ⊥ n
t
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
If two parallel lines are cut by a transversal, then:
(1) corresponding angles are
.
(2) alternate interior angles are
.
(3) same-side interior angles are
.
example: If m ∠1 = 120º, x || y, and m || n, then find all the other measures.
1.
m∠2 =
x
6
4
m∠3 =
5
m∠4 =
y
2
3
1
m
m∠5 =
n
m∠6 =
example: Find the values of x, y, and z.
2.
3.
32º
(3x)º (5y)º
105º
zº
(4x)º
52º
zº
x
(3y+8)º
y
x || y
x=
;y=
;z=
x=
;y=
Assignment: Written Exercises, pages 80 to 82: 1-25 odd #’s
Prepare for Quiz on Lesson 3-2: Properties of Parallel Lines
;z=
Proving Parallel Lines
3-3:
POSTULATE 11
(page 83)
If two lines are cut by a transversal and
are congruent, then the lines are parallel.
angles
t
x
y
THEOREM 3-5
1 2
4 3
5 6
8 7
If two lines are cut by a transversal and
angles are congruent, then the lines are parallel.
interior
Given: transversal t cuts k and n
∠1 ≅ ∠2
Prove: k || n
t
k
3
2
1
n
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
THEOREM 3-6
If two lines are cut by a transversal and
angles are supplementary, then the lines are parallel.
Given: transversal t cuts k and n
∠ 1 is supplementary to ∠ 2
k
Prove: k || n
n
interior
1
2 3
t
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
THEOREM 3-7
In a plane, two lines perpendicular to the
line are parallel.
t
Given: k ⊥ t
k
1
n
2
n ⊥ t
Prove: k || n
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
THEOREM 3-8
Through a point outside a line,there is exactly
the given line.
line parallel to
P
k
THEOREM 3-9
Through a point outside a line,there is exactly
to the given line.
line perpendicular
P
k
THEOREM 3-10
Two lines parallel to a third line are
to each other.
If k || l and k || n,
then __________.
k
NOTE:
l
n
This is true for lines in a plane and for lines in
, however,
Theorems 3-8 and 3-9 are true only for lines in the same
.
Ways to Prove Two Lines Parallel:
(1) Show that a pair of corresponding angles are
.
(2) Show that a pair of alternate interior angles are
.
(3) Show that a pair of same-side interior angles are
(4) In a plane show that both lines are
(5) Show that both lines are
.
to a third line.
to a third line.
example 1: Find the value of “x” that makes a || b, then find the value of “y” that makes m || n.
x=
y=
m
(5x+13)º
n
(6x+1)º
(8y+3)º
a
b
example 2: Given the following information, name the lines (if any) that must be parallel.
(a)
∠1 ≅ ∠8
t
(b)
∠4 ≅ ∠6
(c)
∠10 ≅ ∠5
p
n
k
3
2
4 5
6
1
(d)
∠5 ≅ ∠3
8
(e)
∠2 ≅ ∠7
a
(f)
m∠3+m∠4 = 180º
(g)
m∠5+m∠6 = 180º
(h)
m∠6+m∠9 = 180º
(i)
m∠9+m∠10 = 180º
10
Assignment:Written Exercises, pages 87 & 88: 1-19 odd #’s, 27, 28, 29
Prepare for Quiz on Lessons 3-1 to 3-3: Parallel Lines and Planes
9
7
3-4:
Angles of a Triangle
(page 93)
TRIANGLE: the figure formed by 3 segments joining 3
points.
A
Symbol:
B
VERTEX: each of the
C
points.
Vertices:
;
;
Sides:
;
;
Angles:
;
;
Classifications of Triangles by Sides
Scalene Triangle : no sides are
.
Isosceles Triangle : at least two sides are
.
Equilateral Triangle : all sides are
.
Classifications of Triangles by Angles
Acute Triangle : three
Right Triangle : one
Obtuse Triangle : one
Equiangular Triangle : all angles are
angles.
(90º) angle.
angle.
.
AUXILIARY LINE: a line (ray or segment) added to a diagram to help in a
THEOREM 3-11
.
The sum of the measures of the angles of a triangle is
.
B
Given: ∆ ABC
2
Prove: m∠1 + m∠2 + m∠3 = 180º
1
3
A
C
Proof:
Statements
Reasons
1.
1.
__________________________________
2.
2.
3.
3.
4.
4.
5.
5.
examples: Find the value of “x”.
1.
2.
xº
xº
40º
x=
3.
xº
100º
x=
35º
70º
x=
xº
COROLLARY: a statement that can be
Corollary 1
easily by applying a theorem.
If 2 angles of one triangle are congruent to 2 angles of another
triangle, then the third angles are
.
A
example:
X
Y
B
Corollary 2
Z
C
Each angle of an equiangular triangle has measure
.
example:
Corollary 3
In a triangle, there can be at most
right angle or obtuse angle.
example:
Corollary 4
The acute angles of a right triangle are
.
example:
examples: Find the value of “x”.
4.
5.
50º
40º
xº
xº
40º
x=
xº
x=
xº
EXTERIOR ANGLE (of a triangle): the angle formed when one side of a triangle is
.
example:
REMOTE INTERIOR ANGLES: the angles of the triangle not adjacent to the
angle.
example:
4
1
3
2
THEOREM 3-12
1
4
3
2
The measure of an exterior angle of a triangle equals the
of the measures of the two remote interior angles.
B
2
Given: ∆ABC
Prove: m∠1 + m∠2 = m∠4
1
3
A
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
4
C
examples: Find the value of “x”.
6.
80º
120º
xº
x=
3xº
x=
7.
xº
100º
8.
60º
xº
x=
Assignment: Written Exercises, pages 97 to 99: 1-19 odd #’s, 10, 18, 20, 25, 26, 26
Angles of a Polygon
3-5:
(page 101)
POLYGON: a plane figure formed by
segments (sides) such that:
(1)
each segment intersects exactly
each endpoint; and
(2)
no two segments with a common endpoint are
Polygon means “many
Examples of Polygons
other segments, one at
.
”.
Examples of Figures
that are Not Polygons
A
E
B
D
C
CONVEX POLYGON: a polygon such that no line containing a side of the polygon contains
a point in the
examples:
Nonconvex Polygons - examples:
of the polygon.
DIAGONAL: a segment joining 2 nonvertices of a polygon.
Activity:
Draw all the diagonals from one vertex on each polygon, then complete the chart.
# of sides
of polygon
3
4
5
6
7
8
9
10
11
12
n
name of polygon
total # of
diagonals
from 1 vertex
# of
triangles
formed
sum of angle
measures
THEOREM 3-13
The sum of the measures of the angles of a convex polygon with n sides is
…
.
Activity: Refer to page 104 and do #7.
THEOREM 3-14
The sum of the measures of the exterior angles of any convex polygon,
one angle at each vertex, is
example 1.
º .
If a convex polygon has 24 sides (24-gon), then …
(a) … the interior angle sum is
º.
(b) … and the exterior angle sum is
example 2.
º.
Find the value of “x”.
(a)
(b)
160º
xº
xº
60º
150º
50º
x=
150º
x=
(c)
(d)
120º
60º
140º
60º
xº
xº
150º
x=
160º
xº
x=
xº
REGULAR POLYGON: a polygon that is both
and
.
examples from your template:
Angle Measure of Regular Polygons - for an n-gon the measure of an …
example 3.
example 4.
… interior angle is
.
… exterior angle is
.
Find the measure of each interior angle and each exterior angle of a regular
pentagon.
Each interior angle has measure
.
Each exterior angle has measure
.
How many sides does a regular polygon have if the measure of each exterior
angle is 45º?
The polygon has
example 5.
sides.
How many sides does a regular polygon have if the measure of each interior
angle is 150º?
The polygon has
Assignment: Written Exercises, pages 104 & 105: 8, 10, 11, 13, 15, 16, 17, 19, 20, 21, 22, 25
Prepare for Quiz on Lessons 3-4 & 3-5
sides.
3-6:
Inductive Reasoning
(page 106)
INDUCTIVE REASONING: a kind of reasoning in which the
based on several past observations.
Note:
The conclusion is probably
is
, but not necessarily
.
example 1.
On each of the first 6 days Noah attended his geometry class, Mrs. Heller, his
geometry teacher, gave a homework assignment. Noah concludes that he will
have geometry homework every
he has geometry class.
example 2.
Look for a pattern and predict the next two numbers or letters.
(a)
1, 3, 7, 13, 21,
(b)
81, 27, 9, 3,
(c)
3, -6, 12, -24,
,
,
,
(d)
1, 1, 2, 3, 5, 8, 13, 21,
(e)
O, T, T, F, F, S, S,
(f)
J, M, M, J,
,
,
,
DEDUCTIVE REASONING: proving
by reasoning from accepted
postulates, definitions, theorems, and given information.
Note:
example 3.
The conclusion must be
if hypotheses are
.
In the same geometry class, Hannah reads the theorem “Vertical angles are
congruent.” She notices in a diagram that angle 1 and angle 2 are vertical angles.
Hannah concludes that
.
example 4:
Accept the two statements as given information. State a conclusion based on
deductive reasoning. If no conclusion can be reached, write no conclusion.
(a)
All cows eat grass. Blossom eats grass.
(b)
Aaron is taller than Alex. Alex is taller than Emily.
(c)
∠A ≅ ∠B and m∠A = 72º.
(d)
AB ! CD and AB ! XY .
example 5.
Tell whether the reasoning process is deductive or inductive (circle your answer).
(a) Aaron did his assignment and found the sums of the exterior angles of several different
polygons. Noticing the results were all the same, he concludes that the sum of the
measures of the exterior angles of any polygon is 360 degrees.
deductive or
(b)
Tammy is told that m∠A = 150º and m∠B = 30º. Since she knows the definition of
supplementary angles, she concludes that ∠A and ∠B are supplementary.
deductive or
(c)
inductive
inductive
Nicholas observes that the sum of 2 and 4 is an even number, that the sum of 4 and 6 is
an even number, and that the sum of 12 and 6 is also an even number. He concludes that
the sum of two even numbers is always an even number.
deductive or
Problem:
inductive
Three businessmen stay at a hotel. The hotel room costs $30, therefore, each pays
$10. The owner recalls that they get a discount. The total should be $25. The
owner tells the bellhop to return $5. The bellhop decides to keep $2 and return $1
to each businessman.
Now, each businessman paid $9, totaling $27, plus the $2 the bellhop kept,
totaling $29. Where is the other dollar?
Assignment: Written Exercises, pages 107 & 108: 1-13 ALL #’s, 15-25 odd #’s
Prepare for Test on Chapter 3:Parallel Lines and Planes