Download Postulate 16 Corresponding Angles Converse If 2 lines are cut by a

Lesson 1.1
Patterns and
Inductive Reasoning
You will learn to…
* find and describe patterns
* use inductive reasoning to make
conjectures
Sketch the next figure.
1.
14
2.
2
4
8
16
Describe the pattern.
Find the next three numbers.
3. 0, 1, 1, 2, 3, 5, 8,…
13, 21, 34
4. 4, 9, 16, 25, …
36, 49, 64
5 , __
6 , __
7
5. __
1 , __
2 , __
3 , __
4 , __
2 3 4 5 6 7 8
How many squares are in the next
object?
6.
3,
6,
10,
15
7.
1,
4,
9,
16
A conjecture is an
unproven statement that is
based on a pattern.
Inductive Reasoning is the
process of looking for a
pattern and making a
conjecture.
Complete the conjecture.
8. The sum of any 2 odd
even
numbers is __________.
9. The product of any 2 odd
odd
numbers is __________.
A counterexample is an
example that shows a
conjecture is false.
Find a counterexample.
10. The sum of 2 numbers is
always greater than the
larger of the numbers.
11. If a shape has 2 sides the
same length, then it must be
a rectangle.
12. If a shape has 4 sides, then it
must be square.
Lesson 1.2
Points, Lines,
and Planes
You will learn to…
* understand and use the basic
geometry terms
* sketch intersections of lines and planes
Undefined terms cannot be
mathematically defined using
other known words.
point
line
plane
Two points determine a line.
Postulate 2
Three points determine a plane.
B
A
T
C
plane ABC or plane T
A
B
D
H
E
C
G
F
Do you see…
plane ABF ? plane ADG?
Collinear points
A
points that lie
on the same line
Coplanar points
points that lie
in the same plane
Coplanar lines
lines that lie
in the same plane
C
B
B
A
C
Betweeness refers to
collinear points only.
Point B is between A and C.
A
B
C
Line AB
C
n
A
B
CA
BA
AB
line n
BC
Segment AB
B
A
AB
BA
Is AC the same as AB?
B
A
C
NO
Ray AB
B
A
AB
BA
Opposite Rays share an endpoint
A
NA
N
B
?
NB
a line
Opposite rays form __________.
1) Draw three noncolinear
points J, K, and L.
2) Draw JK, KL, and LJ.
K
J
L
If 2 lines intersect, then their
a point
intersection is ____________.
If 2 planes intersect, then their
a line
intersection is ___________.
If a line intersects a plane they
intersect at __________.
a point
Lesson 1.3
Segments & Their
Measures
You will learn to…
* use segment postulates
* use the distance formula
The distance between points
A and B is written as AB
which is the length of AB.
A
-3
AB =
B
-2 -1 0 1 2 3
| 2
– 3| or |3 –
- 2|
Distance is the absolute
value of their difference.
=5
1. Find the length of the segment.
6
3
inches
=1
1
8
4
Draw a segment that is….
3. 4 cm long
4. 2.7 cm long
5. 56 mm long
A postulate is a statement
or rule that is accepted
without proof.
Rules that are proven are
called theorems.
Segment Addition Postulate
If B is between A and C,
then AB + BC = AC.
AB
A
BC
B
AC
C
6. Find AB.
20
A
B
50
AB + 20 = 50
AB = 30
C
7. Write an expression for AC.
2x - 5
3x + 2
A
B
C
?
AC = (2x – 5) + (3x + 2)
AC = 5x - 3
8. Write an expression for AC.
8x + 1
A
C
E
12x + 10
AC + (8x + 1) = 12x + 10
AC = 4x + 9
9. Suppose M is between L and N. Use
the Segment Addition Postulate to
solve for x.
LM = 3x + 8
L
M
N
MN = 2x – 5
LN = 23
3x + 8 + 2x – 5 = 23
5x + 3 = 23
5x = 20
x=4
10. E is between H and R. A is between
E and R. R is between E and T.
HT = 50, ER = 20, and HE=EA=AR.
Find RT.
50
10
10
10
H E A
R
20
T
EA = AR = 10
RT = 20
HE = 10
11. Find AB and CD.
C
A
B
D
AB= 5
CD= 5
Congruent Segments have
the same lengths.
AB = CD
AB  CD
lengths are equal
segments are congruent
12. Use the Pythagorean Theorem to
find the distance between the points.
(4,6)
2
c=
4
(1,2) 3
2
(4-1)
2
(6-2) ?
+
2
2
(x2-x1) + (y2-y1)
2
c=
2
3
+
2
4
9 + 16
c2= 25
c= 5
Distance Formula
If A (x1, y1) and B (x2, y2) are
points in a coordinate plane, then
AB = (x2 – x1 + (y2 – y1
2
)
2
)
13. Find TM
T (3, -2) and M (1, 4)
TM =
TM =
2
(1 - 3)
2
(-2)
+
+
2
(4 - 2)
2
(6)
TM = 4 + 36 = 40 ≈6.32
4 · 10 = 2 10
Tiled Floor
Street Corner
Two-Point
Perspective
Street Corner
Two-Point
Perspective
Lesson 1.4
Angles and Their
Measures
You will learn to…
* use angle postulates
* classify angles as acute, right, obtuse,
or straight
An angle consists of 2 rays
that have the same endpoint
called the vertex of the angle.
X
Y
YX and YZ
form XYZ
60°
Z
1. Name the sides, vertex, and angle.
Sides: BA and BC
Name:
ABC
Vertex: B
A
CBA
B
B
54°
C
Congruent Angles have the
same measure.
mABC = mXYZ
ABC  XYZ
equal measures
congruent angles
Classifying Angles By Their
Measures…
Acute Angle
Measure is between 0˚ and 90˚
Classifying Angles By Their
Measures…
Obtuse Angle
Measure is between 90˚ and 180˚
Classifying Angles By Their
Measures…
Right Angle
Measure is 90˚
Classifying Angles By Their
Measures…
Straight Angle
Measure is 180˚
Two angles are Adjacent Angles
if they share a vertex and a side
but have
no common interior points.
(not overlapping)
C
A
B
 ABC and
 CBD are
adjacent angles
D
2. List the 3 angles shown.
Which angles are adjacent?
R
P
S
T
 RSP

RSP
 PST

PST
 RST
Angle Addition Postulate
R
P
S
T
m  RSP  m  PST  m  RST
3. Find m  RST.
R
S
35º
94°
P
59º
T
4. Find m  RSP.
60°
110º
R
P
S
50º
T
How do you measure an
angle with a protractor?
5. Use a protractor to draw
a 65˚ angle.
6. Use a protractor to draw
a 112˚ angle.
45° or 55°?
145° or 135°?
45° or 55°?
145° or 135°?
35° or 25°?
155° or 165°?
35° or 25°?
155° or 165°?
120° - 90°
mNPR =
30°
90° - 60°
HW: 29-34
C
B
A
D
E
F
Artist: Julian Beever
People are actually avoiding
walking in the "hole"
'Make Poverty History' drawing from the side (40 ft long)
Lesson 1.5
Segment and Angle
Bisectors
You will learn to…
* bisect a segment
* bisect an angle
A midpoint is a point that
divides a segment into two
congruent segments.
M
A
midpoint
B
AM = MB
AM  MB
To bisect a segment means
to divide it into two
congruent segments.
Use a compass to locate the midpoint of a segment.
A segment bisector intersects
a segment at its midpoint.
How would you find
the “middle” between
2 numbers?
How would you find
the “middle” between
the points?
Midpoint Formula
If A (x1, y1) and B (x2, y2) are
points in a coordinate plane,
xx , y y
2 2
1
2
1
2
1. Find the midpoint
between (-2, 3) and (4, -6)
-2 + 4 , 3 + -6
2
2
(-2,3)
(1, -1.5)
(4,-6)
2. Find the midpoint
between (2, -1) and (4, -4)
2 + 4 , -1 + -4
2
2
(3, -2.5)
3. One endpoint is (-3, -1).
The midpoint is (3, -4).
Find the other endpoint.
(-3,-1), (3,-4), (x,y)
(9, -7)
-3 + x
3
= 31
2
-1 + y
-4
=
-4
1
2
-3 + x = 6
x=9
-1 + y = -8
y = -7
3. One endpoint is (3, -5).
The midpoint is (-2, 4).
Find the other endpoint.
(3,-5), (-2,4), (x,y)
(-7,13)
3+x
= -2
2
3 + x = -4
x = -7
-5 + y 4
=
2
-5 + y = 8
y = 13
An angle bisector is a ray
that divides an angle into
2 congruent angles.
angle bisector
4. NQ is an angle bisector.
Find m MNQ.
mMNQ = 57°
Q
M
114°
N
P
5. NQ is an angle bisector. Find
m
MNQ and m MNP.
mMNQ
=
64°
Q
M
mMNP = 128°
64°
N
P
6. NQ is an angle bisector.
Find x.
2x – 5 = 55
Q
M
x = 30
55°
(2x – 5)°
N
P
7. BD bisects  ABC.
m  ABD = (2x + 50)˚
m  DBC = (5x + 5)˚
Find the measure of all 3 angles.
5x+5 = 2x+50
3x = 45
x = 15
mABC = 160°
A
D
80°
80°
B
C
Workbook
Page 13 (1-8)
Lesson 1.6
2
Angle Pair
Relationships
You will learn to…
* identify vertical angles and linear pairs
* identify complementary and
supplementary angles
Two angles are
Complementary Angles
if the sum of their
measures is 90
A 90˚ angle is a right
angle and forms a corner.
Two angles are
Supplementary Angles
if the sum of their
measures is 180
A 180˚ angle forms a
straight angle.
m  A = 25°
1. Find its complement.
65°
2. Find its supplement.
155°
Do complementary angles have
to be adjacent?
50°
10°
40°
80°
adjacent
nonadjacent
Do supplementary angles have
to be adjacent?
70°
105°
110°
75°
adjacent
nonadjacent
3. Are the two angles formed
complementary or supplementary?
120°
60°
supplementary angles
Two angles are
vertical angles
if they are NOT adjacent
and their sides are formed
by 2 intersecting lines.
vertical angles
1
3
4
2
 1 and  2
 3 and  4
Two angles form a
linear pair
if they are adjacent angles whose
noncommon sides
form a line.
1
3
4
2
linear pairs
 1 and  4
 1 and  3
 2 and  4
 2 and  3
Which angles are…
supplementary? congruent?
 1 and  4
 1 and  3
 2 and  4
 2 and  3
 1 and  2
 3 and  4
1 4
3 2
Linear Pair Postulate
If two angles form a
linear pair, then they are
_____________.
supplementary
Vertical Angles Theorem
Vertical angles are
congruent
___________.
4. Find the measure of
each angle.
m1 = 140°
1
3
40°
2
m2 = 140°
m3 = 40°
5. Find x and y. Then find
the angle measures.
5x+30+4x+15=180
105°
3y-15+3y+15=180
75°
(5x + 30)° (3y - 15)°
(4x + 15)° (3y + 15)°
75°
x = 15
105°
y = 30
Lesson 1.7
Perimeter
& Area
You will learn to…
* find the perimeter and area of
common plane figures
Area
How many squares will
fit inside a region?
Perimeter
What is the distance
around a region?
Square
Area =
A=
2
side
2
s
Perimeter = 4(side)
P = 4s
Rectangle
Area = base ( height )
A = bh
Perimeter = 2(base) + 2(height)
P = 2b + 2h
1. Find the area of a square that
has a side length of 20 inches.
Find the perimeter.
2
20
A=
2
A = 400 inches
P = 4(20)
P = 80 inches
2. Find the area of a rectangle
that is 20 m by 4 m. Find the
perimeter.
A = 20(4)
2
A = 80 m
P = 2(20) + 2(4)
P = 48 m
3. A rectangle has an area of
98 cm2. Find the length of its
base if its height is 7 cm.
A = b(h)
98 = b(7)
b = 14 cm
Triangle
Area = ½ base ( height)
A=½ b h
Perimeter = sum of 3 sides
P= a + b + c
Find the area and perimeter of
the triangle.
P = 40.8 cm
4.
9.8 cm
c2 = 4 2 + 9 2
4
9 cm
15 cm
152 = x2 + 92
12
16 cm
9(16)
2
A=
=
72
cm
2
5. Find the area and perimeter of the
triangle.
A=
9(6)
2
9 cm
?9.8
13.5
?
A = 27 cm2
6 cm
10 cm
P = 29.3 cm
2
c
=
2
9
+
2
2
410
Find the area of the triangle.
6.
25 cm
25(12)
A= 2
12 cm
20 cm
A = 150
2
cm
Find the area of the triangle.
Pythagorean Theorem?
6.
15 cm
25 cm
20(15)
A= 2
12 cm
20 cm
A = 150
2
cm
center
radius
diameter
diameter = 2 (radius)
Circumference of a
Circle
C = (diameter) π
C = (2 · radius) π
C= dπ or C=2rπ
7. Find the circumference.
6 cm
C = 12π cm ≈ 37.7cm
8. Find the circumference.
C = 10π cm ≈ 31.4 cm
10 cm
9.
Find the radius of the circle
that has a circumference of
120 feet.
C = dπC = 2rπC = 2rπ
120 = 2rπ
2π 2π
19.1≈ r
Area of a Circle
2
A = (radius) π
A=
2
r π
10.
Find the area.
6 cm
A = 36π cm2
A ≈ 113.1 cm2
11. Find the area.
A = 64π
2
cm
A ≈ 201.1
2
cm
16 cm
12. Find the radius of the circle that has
an area of 120 square feet.
2
r π
120 =
π
π
2
r
= 38.197
r = 38.197
r ≈ 6.2 ft
13. Find the perimeter and area of the
triangle.
-1)
22
AB = (3
BC
(-2---2)
3)22++(-1
(-4--2)
AC = 6
AB = 34 ≈ 5.8
BC = 34 ≈ 5.8
height = 5
(-2,2)
C
P = 6 + 2 34 ≈ 17.6
A = ½ (5)(6) =
15 square units
B
(3,-1)
A
(-2,-4)
14. Find the area of the bluish region.
Square?
2
A = 12 = 144
12 cm
Circle?
A = 62 π = 36π
Square-Circle = 144 - 36π ≈ 30.9 cm2
15. Find the area of the bluish region.
r=5
Square?
2
A = 20 = 400
Circle?
A = 52 π = 25π
Square-Circles = 400 – 4(25π) ≈ 85.84