Download Chapter 7

Geometry
Chapter 7
Ratios & Proportions
Properties of Proportions
Similar Polygons
Similarity Proofs
Triangle Angle Bisector Theorem
Name: _________________________________________________________________
Geometry
Assignments – Chapter 7
Similar Polygons
Date
Due
Section
Topics
Assignment
Written Exercises
7.1
&
7.2
7.3
& 7.4
7.5




 Similarity
 Similar Polygons
 Scale Factor
AA Similarity





SAS Similarity Postulate
SSS Similarity Postulate
Proportional Lengths
Triangle Proportionality
Theorem
Triangle Angle-Bisector
Theorem
Similarity Proofs

Similarity Proofs
7.6

Similarity
Proofs
Similarity
Proofs
(cont)
&
REVIEW
Ratio
Proportion
Means
Extreme

Pg. 244 # 25-31 odd AND
Pg. 247 #4, 6, 8, 20, 22, 24, 26, 28,
33 & 37
Pg. 250 (bottom)-251 # 2-14 even,
15-22, 24-27 AND
Pg. 257 #2-20 even
Pg. 266 #2-10 even, 14 AND
Pg. 258 # 21-25, 27
Pg. 272-273 #2-11, 20, 22, 23, 25
Worksheet
Worksheet
Chapter Test
Remember, if you have any questions or are having difficulty, please come in for extra help.
1
 x  5   5x  12 
 2 x  3 
92
15
A
105
B
145
12
C
D
2
a c

b d
2 8

3 12
a c

b d
ad  bc
a b

c d
b d

a c
ab cd

b
d
3
4 3
95
6 x
5 x 35

24 9
5.4
6

x
7.2
14 x

x 56
3
7

x  4 x 5
2  5x 2 x  1

8
4
A
AG AH

GB HC
G
B
H
C
4
Ratio and Proportion WS
Geometry – 7-1 and 7-2
Name __________________________
Date _______________ Block ______
Solve each proportion for x.
1.
6 x 27

24 9
2.
4.8 6

x 8.4
3.
4.
6
x

x 150
5.
3
7

x4 x4
6.
7.
10.
x 22

18 12
8
x

x 50
8.
11.
22 2

x 18
3
5

x  3 x 1
2
7
5  10
8
x
5  2 x 3x  1

8
4
5
8
x
9.
1
4
16
12.
3x  7 2 x  1

15
21

5
________________________________________________________________
Complete each proportion. [Use properties!]
w
9
13. Given:
14. Given: 6 : y  h : 7

x
17
w
a)
a) y h 

9
x
6
b)
b)


w
h
6 y
c) 9x 
c)

y
w x
y
d)
d)


x
6
__________________________________________________________________
15. Three numbers aren’t known, but the ratio of the numbers is 1 : 3 : 8.
Is it possible that the numbers are:
a) 1, 3, and 8?
b) 3, 9, and 21?
c) 10, 30, and 80?
d) y, 3y, and 8y?
e) x, 3y, and 8z?
____________________________________________________________________
More practice:
p. 246 Class Exercises (1-12) Challenge: 13, 14
6
GEOMETRY – Notes 7.3
DATE: ____________________________
LOOKING BACK…
Given the statement, “If today is Monday, then I have school.”
Write a. the contrapositive
b. the inverse
c. the converse
Name the quadrilateral:
I have four congruent sides and no right angles. _____________________
I have two congruent segments and the other pair of sides are parallel. ______________
Simplify
18a
36
5x  5 y
x2  y 2
9x  6 y
3
NOTES – 7.3 – SIMILAR POLYGONS
Two polygons are similar if their vertices can be paired so that:
a.
b.
P
D
C
E
Q
T
A
Notation
B
S
R
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the
________________________________.
For the example on the page before, the ________________________ is =
The ratio of the perimeters of two similar polygons is ________________________________
____________________________________________________________________________.
7
Similar Polygons WS1 [notes]
Geometry – section 7-3
Name ______________________
Date _________ Block ______
Determine whether the polygons are similar. Explain your reasoning.
In each question, the given polygons are similar. Find the value of x.
5.
6.
7.
8.
8
In each exercise below, determine whether the polygons are similar. Explain
your reasoning. If the polygons are similar, write a similarity statement.
9.
10.
11.
12.
Complete each of the following.
13.
14.
15. An architect is making plans for a rectangular office building that is 840 feet
long and 252 feet wide. A blueprint of the floor plan for the first floor is 15 inches
long. How wide is the blueprint?
____________________________________________________________________________
9
p.250 CE (1-9)
Are the quadrilaterals similar? If they aren't, tell why not.
1. ABCD and EFGH
2. ABCD and JKLM
3. ABCD and NOPQ
4. JKLM and NOPQ
____________________________________________________________________________
5. If the corresponding angles of two polygons are congruent, must the polygons be similar?
6. If the corresponding sides of two polygons are in proportion, must the polygons be similar?
7. Two polygons are similar. Do they have to be congruent?
8. Two polygons are congruent. Do they have to be similar?
9. Are all regular pentagons similar? Why?
___________________________________________________________
10. JUDY ~ J'U'D'Y'. Complete.
a) m<Y' = _____ and m<D = _____
b) The scale factor of JUDY to J'U'D'Y' is _______
c) Find DU, Y'J' , and J'U'.
d) The ratio of perimeters is ______
e) Explain why it is not true that DUJY ~ Y'J'U'D'.
__________________________________________________________
10
7-4 Notes:
Similarity Proof
Similar Polygons: Corresponding angles of similar polygons are congruent AND
corresponding sides are proportional.
______________________________________________________________________
Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the
two triangles are similar.
Called –
Example: Given :
B I
B
I
A G
C
We can conclude:
H
A
G
11
Algebra first….
12
Similar Triangles WS2
Name __________________________
Geometry – 7-3 & 7-6
Date _____________ Block ________
________________________________________________________________________
Part 1: Use properties of similar triangles to set up equations and solve as
needed. Show your work!
1.
2.
3.
4.
13
5.
6.
7.
________________________________________________________________________
Part 2: Proving the Triangle Proportionality Theorem. Fill in the following proof…
Statements
1) DE BC
2)
ADE 
3)
ABC
4)
Reasons
given
ABC and
AED 
ADE
.
ACB
AA similarity
AB CA

DA EA
5) AD + DB = AB; AE + EC = AC
6)
AD  DB AE  EC

DA
EA
7)
BD CE

DA EA
 If a line parallel to one side of a triangle intersects the other two sides, then it
divides those sides proportionally.
14
________________________________________________________________________
Part 3: Use similar triangles or the new theorem from part 2 (triangle
proportionality theorem) to solve for the value of x in each question.
15
2)
Given: <QVR ≅ <S
Prove: (QR)(ST) = (QT)(VR)
16
17
p.271 CE (3-5). Write a proportion for each question. Solve question 4 & 5.
p.272 WE(6-8). Find the value of x.
18
7-4 & 7-5 Triangle Similarity Proofs
AA Similarity Theorem:
SSS Similarity Theorem:
SAS Similarity Theorem:
_____________________________________________________________________________
Similarity Proofs WS1
Name _____________________________
Geometry
Date _________ Block ___________
B
A
1. Given: AB DC
Prove: AE  DE  CE  BE
E
D
2. Given: ABCD is a parallelogram
Prove: AF  EF  DF  BF
C
B
C
A
D
F
E
3. Given: ABC; D midpoint of AC; E midpoint of BC
CD DE
Prove:

CA
AB
[hint : segment DE must be a midsegment…]
A
C
D
E
B
19
4. Given:
ABC with AB  AC; PD  AB;
A
AE  BC
EC AC
Prove:

DB PB
D
B
C
E
________________________________________________________________________
C
5. Find the value of x.
given: BC  CA; CD  BA
and lengths in diagram
x
[use similar triangles]
B
6
A
24
D
P
6. If the ratio of the lengths of the segments formed on a hypotenuse of a right
triangle from the intersection of the altitude is 1:9; and the length of the altitude
to the hypotenuse is 6, find the lengths of the two segments formed on the
hypotenuse. [use similar triangles]
7. Find x.
given: BD  AC; AB  BC; AC  50
and lengths in diagram
[use similar triangles]
A
x
D
30
B
8. Find x.
given: the two horizontal segments
are parallel; not to scale!
C
6
10
x
15
9. Find x.
given: the three vertical segments are
parallel
x+1
2
x
x+6
20
Similarity Proof WS2
Geometry
Name _________________________
Date _____________ Block ______
______________________________________________________________________
21
__________________________________________________________________
22
______________________________________________________________________
23
Similarity Proofs WS3
Geometry
Name ____________________________
Date ________________ Block ______
1. Given: A  C
2. Given:
Prove:
ABX
VZ WZ

XZ YZ
Prove: 1  2
CDX
B
D
Z
W
V
X
X
Y
C
A
BA  BC ;
3. Given: PST  PRQ
4. Given: AE  BC ;
PS PT
Prove:

PR PQ
BD  AC
AC AE
Prove:

BA BD
B
P
S
E
T
R
Q
A
5. Given: EDA  DAC
Prove: BED
BAC
6. Given: 1  A
Prove: ABE
CDE
D
7. Given: 1  2
Prove: MLN
JLK
B
L
1
B
E
A
C
D
M
E
C
C
A
J
1
N
2
K
D
24
Chapter 7 Review
Geometry
Name __________________________
Date __________ Block _______
1. Two angles are complementary and are in the ratio of 7:8. Find the value of the
smaller angle.
2. Quad ABCD with m<A: m<B: m<C : m<D = 7:2:2:7. Find the measure of all the
angles in the quadrilateral and state what special type of quadrilateral ABCD is.
3. A hexagons angles are in the ratio of 4 : 3 : 7 : 3 : 6 : 7 , find the measure of
the largest angle.
4. Solve the following proportions for all variables.
a.
9
5

3x  6 x  3
b.
12
x

x4 5
d.
4 x

x 9
e.
y  x 3 x
x5 y
and


3
2
4
5
c.
x  7 x  13

2
x2
5. Determine whether the given figures are sometimes, always or never similar.
a. Two Rectangles
b. Two Squares
c. Two isosceles trapezoids with congruent base angles
d. Two regular dodecagons
e. A scalene triangle and an isosceles triangle
f. Two rhombi with at least one < congruent
g. Two triangles with proportional sides
6. Quad ABCD ~ Quad HIJK
a. If, AB = 8, JK = 12, and CD = 9, find HI.
b. If m<B = 50 , find m<I.
c. If the perimeter of Quad HIJK is equal to 36, what is the perimeter
of Quad ABCD?
7. If 2500 square feet of grass emits enough oxygen for a family of 4, how much
grass is needed to supply oxygen for a family of five?
25
8. If a car uses 15 gallons of gas to travel 500 miles, how many miles does it get
for one gallon of gas?
9. Prove whether or not the following triangles are similar. Justify your answer.
15
5
4
10. Find x.
25
20
3
11. Find w, x, y, z.
[diagram not to scale!]
12. Use the diagram below to answer each question. Segment AC is an angle
bisector.
a. If BC = 20, AB = 18, AD = 45 find CD.
b. If AB = 12, AD = 15, and BD = 9, find CD.
13. Given: AC  AB , BV  AB
Prove: AC JB  BV CJ
14. Given: Rectangle ABCD ,  EFB   CGE
Prove
ABG ~ DCF
26
15.
16.
17.
18.
Given: Iso. Trap. ABFC with AC  BF , AD // BF
Prove: AD  DF  EF  ED
19. Given:
Prove:
1 2
A D
27
EXTRA PRACTICE – See me for answers:
Pg 252 (Self Test #1) #1-6
Pg 274 (Self Test #2) #1-11
Pg 277-278 (Chpt Rev) #1-24
Pg 279 (Chpt Test) #1-15
28