Transcript Math 7 Unit 4 G E O

GEOMETRY
Math 7 Unit 4
Standards
Strand 4:
Concept 1
PO 1. Draw a model that demonstrates basic geometric relationships such
as parallelism, perpendicularity, similarity/proportionality, and
congruence.
PO 6. Identify the properties of angles created by a transversal
intersecting two parallel lines
PO 7. Recognize the relationship between inscribed angles and
intercepted arcs.
PO 8. Identify tangents and secants of a circle.
PO 9. Determine whether three given lengths can form a triangle.
PO 10. Identify corresponding angles of similar polygons as congruent
and sides as proportional.
Strand 4:
Concept 4
PO 6. Solve problems using ratios and proportions, given the scale factor.
PO 7. Calculate the length of a side given two similar triangles.
IN FLAGS
IN NATURE
IN SPORTS
IN MUSIC
IN SCIENCE
IN Games
IN BUILDINGS
The hardest part about Geometry
Point
A
: a location in space
: think about the tip of your pencil
●A
Line
A
B
all the points on a never-ending
straight path that extends in all
directions

AB
Segment
C
D
all the points on a straight path
between 2 points, including those
endpoints
CD
Ray
E
F
a part of a line that starts at a
point (endpoint) and extends
forever in one direction

EF
Angle
formed by 2 rays that
share the same
endpoint. The point is
called the VERTEX and
the rays are called the
sides. Angles are
measure in degrees.
70
Side
A
1
B
C
Side
Vertex
Angle
A
ABC
B
1
15°
B
C
Plane
a flat surface without thickness
extending in all directions
Think: a wall, a floor, a sheet of paper
A
Parallel Lines
A
B
C
D
lines that never intersect (meet)
and are the same distance apart

AB
║

CD
Perpendicular Lines
D
A
B
C
lines that meet to form right angles


AB  CD
Intersecting Lines
D
A
B
C
lines that meet at a point
Right Angle
An angle that
measures
90 degrees.
Straight Angle
An angle that measures
180 degrees or 0.
(straight line)
Acute Angle
An angle that measures
between 1 and 89 degrees
Obtuse Angle
An angle that measures
between 91 and 179
degrees
Complementary Angles
Two or more
angles whose
measures total 90
degrees.
2
1
Supplementary Angles
Two or more
angles that
add up to
180 degrees.
1
2
*****Reminders******
Supplementary
Straight angle
Complimentary
Corner
Adjacent Angles
A
D
B
C
Two angles who share a
common side
Example 1
• Estimate the measure of the angle, then use
a protractor to find the measure of the angle.
Example 1
• Angles 1 and 2 are
complementary. If
• m  1 = 60,
find m  2.
1
+ 2 = 90
2
= 90 -  1
2
2
1
60
= 90 - 60
 2 = 30
Example 3
• Angles 1 and 2 are supplementary.
If m  1 is 114, find m 2.
< 1 + < 2 = 180
< 2 = 180 - < 1
< 2 = 180 - 114
< 2 = 66
2
1
7.2 Angle Relationships
t
1
4
3
6
5
7
2
8
Vertical Angles
• Two angles that are opposite angles.
• Vertical angels are always congruent!
1   3
2   4
Vertical Angles
• Example 1: Find the measures of the
t
missing angles
125 
?
125 
?
55 
PARALLEL LINES
• Def: line that do not intersect.
• Illustration:
l
B
A
D
C
m
l
|| m
AB || CD
Examples of Parallel Lines
•
•
•
•
•
Hardwood Floor
Opposite sides of windows, desks, etc.
Parking slots in parking lot
Parallel Parking
Streets: Arizona Avenue and Alma School Rd.
Examples of Parallel Lines
• Streets: Belmont & School
Transversal
• Def: a line that intersects two lines
at different points
t
• Illustration:
Supplementary Angles/
Linear Pair
• Two angles that form a line (sum=180)
1+2=180
2+4=180
4+3=180
3+1=180
t
1 2
3 4
5
7
6
8
5+6=180
6+8=180
8+7=180
7+5=180
Supplementary Angles/
Linear Pair
• Find the measures of the missing
t
angles
108? 72 
108
? 
Alternate Exterior Angles
• Two angles that lie outside parallel lines
on opposite sides of the transversal
t
1
2
3
4
5 6
7
8
2   7
1   8
Alternate Interior Angles
• Two angles that lie between parallel
lines on opposite sides of the
transversal t
1
2
3
4
5 6
7
8
3   6
4   5
Corresponding Angles
• Two angles that occupy corresponding
positions. t
Top Left
Top Right
1
3
Bottom Left
Top Left
Bottom Left
2
4
Bottom Right
5
6 Top Right
7
8
Bottom Right
1   5
2   6
3   7
4   8
Same Side Interior Angles
3 +5 = 180
4 +6 = 180
t
1
2
3
4
5 6
7
8
• Two angles that
lie between
parallel lines on
the same sides
of the
transversal
List all pairs of angles that fit the
description.
5
a.
b.
c.
d.
Corresponding
Alternate Interior
Alternate Exterior
Consecutive Interior
4
6
3
7
2
1
8
t
Find all angle measures
t
180 - 67
1131 67 
672
3113 
1135 67
8 
67 6
 113
7 
Example 5:
• find the m 1, if m 3 = 57
• find m 4, if m 5 = 136
• find the m 2, if m 7 = 84
Algebraic Angles
• Name the angle relationship
= 90
– Are they congruent, complementary or
supplementary?
– Complementary
x + 36 = 90
• Find the value of x
-36
-36
x = 54
36 
x 
Example 2
• Name the angle relationship
– Vertical
– Are they congruent, complementary or
supplementary?
• Find the value of x

x = 115 
115 
x
Example 3
• Name the angle relationship
– Alternate Exterior
– Are they congruent, complementary or
supplementary?
• Find the value of x
t
125 
5x 
5x = 125
5
5
x = 25

Example 4
• Name the angle relationship
– Corresponding
– Are they congruent, complementary or
supplementary?
• Find the value of x
t
2x + 1
151

2x + 1 = 151
-1
-1
2x = 150
2
2
x = 75
Example 5
• Name the angle relationship
– Consecutive Interior Angles
– Are they congruent, complementary or
supplementary?
• Find the value of x
7x + 15 + 81
t
81
7x + 15
supp
= 180
7x + 96 = 180
- 96 - 96
7x = 84
7
7
x = 12
Example 6
• Name the angle relationship
– Alternate Interior Angles
– Are they congruent, complementary or
supplementary?
• Find the value of x
t
3x
2x + 20
2x + 20 = 3x
- 2x
- 2x
20 = x

The World Of Triangles
Pick Up Sticks
• For each given set of rods, determine if the
rods can be placed together to form a
triangle. In order to count as a triangle,
every rod must be touching corner to corner.
See example below.
Colors
Does it make a triangle?
Y/N
a. orange, blue dark green
Yes
b. light green, yellow, dark green
c. red, white, black
Yes
No
d. yellow, brown, light green
No
e. dark green, yellow, red
Yes
f. purple, dark green, white
No
g. orange, blue, white
No
h. black, dark green, red
Yes
• Can you use two of the same color rods and
make a triangle? Explain and give an
example.
Now find five new sets of three rods that can
form a triangle. Find five new sets of rods that
will not make a triangle.
Makes a triangle
Does not make a
triangle
Without actually putting them
together, how can you tell whether
or not three rods will form a triangle?
Triangles
• A triangle is a 3-sided
polygon. Every
triangle has three sides
and three angles,
which when added
together equal 180°.
Triangle Inequality:
• In order for three sides to form a triangle,
the sum of the two smaller sides must be
greater than the largest.
Triangle Inequality:
Examples:
Can the following sides form a triangle? Why or Why not?
A.
1,2,2
B. 5,6,15
Given the lengths of two sides of a triangle, state
the greatest whole-number measurement that is
possible for the third.
A.
3,5
B. 2,8
TRIANGLES
Triangles can be classified
according to the size of
their angles.
Right Triangles
• A right triangle is
triangle with an
angle of 90°.
Obtuse Triangles
• An obtuse
triangle is a
triangle in which
one of the angles
is greater than
90°.
Acute Triangles
• A triangle in
which all three
angles are less
than 90°.
Triangles
Triangles can be classified
according to the length of their
sides.
Scalene Triangles
• A triangle with
three unequal
sides.
Isosceles Triangles
• An isosceles
triangle is a
triangle with two
equal sides.
Equilateral Triangles
• An equilateral
triangle is a triangle
with all three sides
of equal length.
• Equilateral
triangles are also
equilangular. (all
angles the same)
The sum of the interior angles of
a triangle is 180 degrees.
• Examples: Find the missing angle:
x
42
50
70
x
The sum of the interior angles of
a quadrilateral is 360 degrees.
• Examples: Find the missing angle:
80
80
x
60
x
7.5 NOTES Congruent and Similar
• Def’n - congruent – In geometry, figures are
congruent when they are exactly the same
size and shape.
• Congruent figures have corresponding sides
and angles that are equal.
BC  _______
Symbol:
D
B
C
A
EX. 1
E
F
AB  _______
ABC  _______
BC  _______
________  EFD
_________  EF
CAB  _______
All corresponding parts are congruent so
ABC

EDF
Similar
•
•
Def’n – similar – Figures that have the
same shape but differ in size are similar.
Corresponding angles are equal.
Symbol: ~
Example 2
D
A
E
B
F
C
________________ ~ _________________
Example 3: Find the value of x
in each pair of figures.
R
16 ft
O
M
E
H
S
L
B
T
J
62 in
2x ft
S
ROB  STL
• Corresponding
sides are equal so
2x = 16
2
2
x = 8 ft
I
K
3x + 32 in
MIKE
O
JOSH
3x + 32 = 62
-32 -32
3x = 30
3
3
x = 10 in
Example 4
• Sketch both triangles and properly label each
vertex. Then list the three pairs of sides and
three pairs of angles that are congruent.
NOTES on Similar Figures/Indirect
Measurement
• Recall that similar figures have corresponding
angles that are CONGRUENT but their sides are
PROPORTIONAL.
• Def’n – ratio of the corresponding side lengths
of similar figures (a.k.a. SCALE FACTOR) –
corresponding sides of congruent triangles are
proportional. One side of the first triangle over
the matching side on the second triangle.
EX. 1 The triangles below are similar.
S
6
a) Find the ratio of the
corresponding side lengths. 5
b) Complete each statement.
ST
i.) RT
ii.) RST
UW

VW
R
105
6 in.
~
UVW
c) Find the measure of <VWU.
105
iii.)
T
U 5 in. W
mVUW
mSRT
V
EX. 2 Write a mathematical statement saying the
figures are similar.
Show which angles and sides correspond.
A
D
B
C
H
K
I
J
ABCD~
IHKJ
You can use similar triangles to find the
measure of objects we can’t measure.
• Use a proportion to solve for x.
• Example If ABC ~ DEF
find the value of x.
30x = 240
30
30
x = 8 ft
Example 2:
7 x

5 10
5x = 70
5
5
x = 14 mm
Example 3: A basketball pole is 10 feet high and
casts a shadow of 12 feet. A girl standing nearby is 5
feet tall. How long is the shadow that she casts?
10 12

5
x
10x = 60
10
10
x = 6 ft
Example 4: Use similar triangles to
find the distance across the pond.
8 10

x 45
10x = 360
10
10
x = 36 m
CIRCLES
Radius (or Radii for plural)
A
• The segment
joining the center
of a circle to a
point on the circle.
O
Example: OA
Chord
B
A
A segment joining
two points on a
circle
C
Example: AB
Diameter
• A chord that
passes through the
center of a circle.
A
O
Example: AB
B
Secant
A line that intersects
the circle at
exactly two points.
A
C
O
Example: AB
D
B
Tangent
B
C
• A line that intersects
a circle at exactly
one point.
A
Example: AB
Arc
• A figure consisting of
two points on a circle
and all the points on
the circle needed to
connect them by a
single path.
(
Example: AB
B
A
Central Angle
• An angle whose
vertex is at the
center of a circle.
G
Q
Example: <GQH
H
Inscribed Angle
M
N
T
• An angle whose
vertex is on a
circle and whose
sides are
determined by two
chords.
Example: <MTN
Intercepted Arc
M
N
• An arc that lies in
the interior of an
inscribed angle.
(
Example: MN
T
Important Information
M
N
An inscribed angle is equal
in measure to half of the
measure of its intercepted
arc.
(
So the measure < MTN
of is equal to ½ of the
measure of MN
T
EX. 1 Refer to the picture
at the right.
a) Name a tangent: EF
b) Name a secant:
c) Name a chord:
BD
B
E
A
AD
C
27
d) Name an inscribed angle:
<ADB
e) Give the measure of arc AB.
54
F
D