“Ratios and Proportions”
Ratio—compares two quantities in a fraction form
with one number over another number.
a
b
Proportion—two equal ratios.
a c

b d
To Solve Proportions
“Cross-Multiply”
a c

b d
ad = bc
Multiply across from upper left to
lower right and from upper right
to lower left.
x 5

y 8
What do you get?
8x = 5y
Ex #1:
Solve for x in these Proportions
x 2

35 5
“Cross-Multiply”
5x= 70
x = 14
x x 1

2
3
“Cross-Multiply”
3x = 2(x + 1)
3x = 2x + 2
1x = 2
x=2
Ex #2:
Solve for x in this Proportion
x  7 x 5

2
5
“Cross Multiply”
5(x – 7) = 2(x + 5)
5x – 35 = 2x + 10
3x – 35 = 10
3x = 45
x = 15
STUDY FOR UNIT 4 TEST
(Friday , Dec. 14th or Monday, Dec. 17th)


Triangles

Congruent Polygons

Congruent Triangles

SSS, SAS, ASA, AAS Problems

SSS, SAS, ASA, AAS Proofs

CPCTC and CPCTC Proofs

HL Theorem
Equilateral, Isosceles, and Right Triangle Problems and Solving for
Missing Variables
Problem of the Day
Return Test and Collect SOL HW #5
“Angle 1 and Angle 2 are Supplementary Angles.
m<1 = 12x + 8 and m<2 = 8x + 12. Find the m<2?”
A. 20 degrees
B. 28 degrees
C. 104 degrees
D. 76 degrees
Section 7.2 “Similar Polygons”
Ex I.:
Ratio Problem
A Scale Model of a car is 4 in long. The actual car is 180
in long. What is the ratio of the length of the model to
the length of the car?
Model Car = 4 in
Real Car 180 in
1 in
45 in
Similar (~) and 1 Note
Two figures that have the same shape but not
necessarily the same size are similar.
1. When solving Similar Problems, use Proportions.
To Tell if Similar Polygons??
1.
Corresponding Angles
must be Equal.
2. Corresponding Sides must
be in Proportional Ratios.
T
Triangle ABC ~ RST
B
50
10
20
85
45
A
22
C
11
45 85 R
10
50 5
S
Ex #1:
Complete each Statement
ABCD ~ EFGH
B
C
I.
II.
III.
A
53
D
IV.
F 127
E
G
H
V.
m<E =
m<B =
AB = AD
EF
?
BC = AB
? EF
GH = FG
CD ?
Try Example #2
I.
II.
III.
IV.
V.
<H
<R
<X
HX
HR
Ex #3:
I. Polygons Similar (Yes/No)?
II. Why w/Angles and Sides?
I.
B
15
12
C
A
E
18
16
D
20
24
II. Angles Equal
10
A
E
18
F
D
B
14
C
15
12
20
F
Try Example #4
No, Sides are Not Proportional Ratios
II. Yes, Angles Equal and Sides are ¾ Ratio
I.
Ex #5:
For both, Find x and y w/Similar Figures
LMNO ~ QRST
T
O 2 N
Q
ABC ~ YWZ
x
6
M
S
40
R
15
L
Y
A
y
B
x
12
20
W
C
Z
30
Try Example #6
x = 6 and y = 5
II. x = 9
I.
Worksheet “Similar Polygons”
Problem of the Day #1
**Check Homework**
Yes, by 1/2 Ratio
2. No, Angles are not Equal
3. x = 4
4. x = 20 and y = 8
1.
Problem of the Day #2
**Show Grades or Show Homework/Return Work**
1.
2.
x = 5.3 and y = 36
I. x = 7
II. BC = 11
III. MN = 22
“Proving Triangles Similar”
Try—3pts + 2pt EC Correct
Find the Height of the Tree?
2m
3m
30m
Example #1
“A Small Child is 3 feet tall and is standing 6 feet from a
flag pole. Another person is standing 12 feet from the
same flag pole. How tall is that other person?”
[Draw Picture]
3 Similarities
1. Angle-Angle (AA~) Similarity
“If Two Angles Congruent, then Triangles are Similar.”
2. Side-Angle-Side (SAS~) Similarity
“If Two Sides are Proportional and the middle Angle is
Congruent, then Triangles are Similar.”
3. Side-Side-Side (SSS~) Similarity
“If all three Sides are Proportional, then Triangles are Similar.”
One More Note
Little Triangle Inside Big Triangle, then AA~.
Ex #2a:
Explain why these Triangles are Similar?
Write a Similarity Statement?
W
T
70
R
70
M
L
<WMR = <TML (Vertical Angles), so
Triangle WMR ~ Triangle TML by AA~
Ex #2b:
Explain why these Triangles are Similar?
Write a Similarity Statement?
G
R
24
F
16
H
<F = <M are Right Angles
12/16 = ¾
18/24 = ¾
12
M
18
K
Triangle FGH ~ Triangle MKR by SAS~
Ex #2c:
Explain why these Triangles are Similar?
Write a Similarity Statement?
H
10
T
10
G
12
C
25
12
N
M
30
10/12 = 5/6
10/12 = 5/6
25/30 = 5/6
Triangle GHC ~ Triangle NTM by SSS~
Ex #3:
Explain why Similar? Then, find x?
A
10
B
12
C
X
16
D
18
X
4
12
E
SSS~ Postulate
12 = 10
18 X
12X = 180
X = 15
SAS~ Postulate
X=4
12 16
16X = 48
X=3
Try Examples #4
AA~ or SAS~ and x = 9
II. SAS~ and x = 90
I.
Ex #5:
Explain why Similar? Then, find x?
4
6
9
2
X
AA~ Postulate
6=6+2
8
9
X
6X = 72
X = 12
5
X
15
AA~ Postulate
4=X+4
5
15
5X + 20 = 60; 5X = 40
X=8
EXIT SLIP Worth: +10 Points
**COLLECT**
Explain why Similar? Then, find x?
I.
II.
A
8
B
4
C
D
6
90
X
110
E
90
X
1. Worksheet “Similarity in Triangles”
2. SHORT QUIZ (Next Block)
 Similar
Polygons
 Similar Polygon Problems
 Similarity in Triangles (AA~, SAS~, and SSS~)
 Similarity in Triangle Solving Problems
Problem of the Day
**Check Worksheet and Then Quiz**
I. Explain why Similar (AA~, SAS~, SSS~)?
II. Find x?
3
7
9
X
Then New Notes
“Proportions in Triangles”
‘2’ Theorems
1. Triangle-Angle-Bisector Theorem
“If an angle bisector bisects a triangle, then it divides the opposite
side into two segments that are proportional to the triangle sides.”
A
B
C
D
AC CD

AB BD
Ex #1:
Find x w/Triangle-Angle-Bisector
A
6
8
X
B
5
D 5
C
x
8
AC CD

AB BD
3
x 8

6 5
5x = 48
x = 9.6
5 8

3 x
5x = 24
x = 4.8
Ex #2:
Find x w/Triangle-Angle-Bisector
A
4
6
8
B
6
D
C
x
x
6
x = 3.6
10
x = 5.7
2. Side-Splitter Theorem
“If a line is parallel to one side of a triangle and intersects the other
two sides, then it divides those sides in proportions.”
A
B
C
AB AC

BD CE
c
a
b
D
E
d
a c

b d
Ex #3:
Find x w/Side-Splitter Theorem
I.
T
x
S
16
R
II.
5
x + 10
U
10
V
TS TU

SR UV Plug in values:
x
5

16 10
x
x  10 5

x
3
10x = 80
3x + 30 = 5x
30 = 2x
x=8
15 = x
5
3
Try Ex #4:
Find x and y?
6
x
9
7
14
y
X = 12 and Y = 10.5
+ 2pt EC Correct
Solve
“A man who is 6 feet tall casts a shadow that is 4 feet
long. At the same time, a nearby flagpole casts a
shadow that is 14 feet long. How tall is the flagpole?”
1. Worksheets “2 Theorems”
2. Test—Unit 5
(Friday, January 11th 1st, 5th, 7th Blocks
Monday, January 14th 2nd and 6th Blocks)
 Ratio and
Proportion Problems
 Scale Model w/Ratio Problems
 Cross-Multiply to Find “x” Problems
 Similar Polygons and their Problems
 Similarity in Triangles (AA~, SAS~, and SSS~)
 Similarity in Triangle Solving Problems
 Triangle Angle Bisector and Side-Splitter Theorems
 Triangle Inequality Problems
Problem of the Day
Find x for Both?
**Check Worksheet**
4
X
x = 12
2
6
x+6
x–2
X = 4.7
8
2
Inequalities in Triangles
‘3’ Theorems
st
1
‘2’ Theorems
If an Unequal , (1) the longest side is across from the
largest angle and (2) the largest angle is across from
the longest side.
If <A is largest, then BC is the longest side.
B
C
A
Ex #1:
List the Angles in Size from Smallest to Largest
K
21 ft
38 ft
(Greater then)
L
36 ft
Since Side KL is the smallest,
<M then <K then <L
M
Ex #2:
From a Triangle, List Angles from biggest to
smallest
In Triangle ABC with
AB = 12 ft
AC = 11 ft
BC = 8 ft
[Hint: Draw the Triangle]
Biggest to Smallest Angles
<C to <B to <A
Ex #3:
1. Find x & 2. Find the Shortest and Longest Sides?
I.
N
x
II. H
x
64
82
C
B
(Greater then)
57
32
K
V
x = 34 degrees
Shortest Side = CV
Longest Side = NC
x = 91 degrees
Shortest Side = BH
Longest Side = BK
Ex #4:
From a Triangle, List Sides from shortest to longest
In Triangle ABC with
<A = 90 degrees
<B = 40 degrees
<C = 50 degrees
[Hint: Draw the Triangle]
Short to Long Sides
AC, AB, to BC
rd
3
Theorem
w/Wooden Popsicle Stick Activity (+8pts)
Cut the first Popsicle Stick into 1 in., 1 in., and 4 in. pieces.
2. Try to make a Triangle out of these three cut-out pieces. Glue
them down on a piece of paper.
3. Cut the second Popsicle Stick into 2 in., 2 in., and 2 in. pieces.
4. Try to make a Triangle out of these three cut-out pieces. Glue
them down on a piece of paper.
Answer these Questions:
1. Which makes a Triangle (1st or 2nd Popsicle Stick)?
2. Why??
1.
Inequality Theorem:
If the sum of the lengths of a triangle is greater then the third side, then YES a
triangle. If not, NO a triangle.
Ex #5: For all Three
A Triangle (YES/NO)?
I.
Sides 3 ft, 7ft, 8ft
II.
Sides 3cm, 6cm, 1ocm
3 + 6 > 10 (No)
3 + 7 > 8 (Yes)
7 + 8 > 3 (Yes)
3 + 8 > 7 (Yes)
NO, not a Triangle
YES, a Triangle
III. Sides 1 ft, 9 ft, and 9ft
YES, a Triangle
Example #6
Which of these three lengths COULD NOT be the
lengths of the sides of a triangle? WHY??
A 7 m, 9 m, 5 m
B 3 m, 6 m, 9 m
C 5 m, 7 m, 8 m
Try Example #7
Which of these three lengths can COULD be the
lengths of the sides of a triangle? WHY??
A 3 m, 14 m, 17 m
B 11 m, 8 m, 12 m
C 2 m, 3 m, 7 m
Example #8
Find the Possible Length of the Third Side, TK?
T
35 ft
(Greater then)
K
A.
B.
C.
D.
10 < x < 45
35 < x < 80
10 < x < 80
10 < x < 80
45 ft
H
Try Example #9
Find the Possible Length of the Third Side, TK?
T
25 ft
(Greater then)
K
A.
B.
C.
D.
25 < x < 65
15 < x < 40
15 < x < 65
15 < x < 65
40 ft
H
Worksheets “Triangle Inequalities”
2. Test—Unit 5
(Friday, January 11th 1st, 5th, 7th Blocks
Monday, January 14th 2nd and 6th Blocks)
1.
 Ratio and
Proportion Problems
 Scale Model w/Ratio Problems
 Cross-Multiply to Find “x” Problems
 Similar Polygons and their Problems
 Similarity in Triangles (AA~, SAS~, and SSS~)
 Similarity in Triangle Solving Problems
 Triangle Angle Bisector and Side-Splitter Theorems
 Triangle Inequality Problems
Problem of the Day**Check HW and Then Test**
SOL TEI Review Question
120
105
30
130
75
Worth: +10pts
1. A
2. B
3. C
4. C
5. D
6. B
7. A
8. A
9. x = 80 degrees and BD (Longest Side)
80
SOL Homework #6
2. Semester Review Booklet
3. Extra Credit Sheet
1.