Name: ____________________________________ Block:________________

Chapter 6
Similarity

Sections Covered:
6.1 Ratios, Proportions, and Geometric Mean
6.2 Proportions in Geometric Problem Solving
6.3 Similar Polygons
6.4 Prove Triangles Similar by AA
6.5 Prove Triangles Similar by SSS and SAS
6.6 Proportionality Theorems

The student, given information in the form of a figure or statement, will

SOL G.7 prove two triangles are similar, using algebraic and coordinate methods
as well as deductive proofs.

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Unit 5 Syllabus: Ch. 6 Similarity

Block Date Topic Homework

11 M 12/8 6.1 Ratios, Proportions and Geometric Worksheet 6.2 and 6.3
T 12/9 Mean
6.2 Use Proportions to Solve Geometry
Problems
6.3 Use Similar Polygons
12 W 12/10 6.4 Prove Triangles Similar by AA Worksheet 6.4 and 6.5
Th 12/11 6.5 Prove Triangles Similar by SSS and
SAS
13 F 12/12 6.6 Use Proportionality Theorems Worksheet 6.6
M 12/15
14 T 12/16 Ch 6 Review Worksheet Ch 6 Review
W 12/17
15 Th 12/18 Ch 6 Quest Mid-Term Review Packet
F 12/19

***Syllabus subject to change due to weather, pep rallies, illness, etc

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and Monday, Wednesday, and Thursday afternoons.

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____________________________________________

Ratio of a to b  a  a:b
b

1 m = 100 cm 12 in = 1 ft
Simplifying Ratios:  denominators cannot be zero 16 oz= 1 lb 3 ft = 1 yd
 must have the same units 5, 280 ft = 1 mi
 must be simplified 1,760 yd = 1 mi

12cm 6ft 24oz 14ft 440yd

1. 2. 3. 4. 5.
4m 18in 2lb 6yd 2mi

6. The area of a rectangle is 108 cm2. 7. If the measures of the angles in a triangle
The ratio of the width to the length is 3:4. have the ratio of 4:5:6, classify the triangle
Find the length and the width. as right, obtuse or acute.

BD BE DE BE
8. In the diagram, = . Find BA and BD. 9. In the diagram, = . Find AC.
DA EC AC BC

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 The ____________________ of two positive numbers a and b is
Geometric the positive number x that satisfies:
Mean = so x 2 = ab

11. Find the Geometric Mean of:

a. 4 and 9 b. 16 and 18 c. 6 and 20 d. 8 is the geometric mean of 4 and what number?

Notes: 6.2 Use Proportions to Solve Geometry Problems

____________________________________________

a c
Proportion: equation that equates two ratios =
b d
Properties:
a. Cross Products b. Reciprocal
a c a c
If = , then If = , then
b d b d

a c a a c a+b
c. If = , then = . d. If = , then = .
b d c b d b

Practice: Complete each statement.

6 5 6 x y x
1. If = , then = . 2. If = , then = .
x y 5 12 26 y

x 7 x+4 9 x 11
3. If = , then = . 4. If = , then = .
4 y 4 2 y 2

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 Decide whether the statement is True or False.
x 8 y 3 x 8 3 y x 8 x 3
5. If = , then = . 6. If = , then = . 7. If = , then = .
y 3 x 8 y 3 x 8 y 3 8 y

x 8 x y x 8 x+8 y+3
8. If = , then = . 9. If = , then = .
y 3 8 3 y 3 8 3

Solve for x.
x 9 4 3 5 3
10. = 11. = 12. =
6 24 y +3 y -3 2y - 7 y - 3

Solve for the variable.

13. MN:MO is 3:4 14. PQR side lengths: STU side lengths is 1:3
S P
M
x 5
x 9
R Q
O 12
U
N 36 T

Use the diagram and the given information to find the unknown length.
AB AE AB AE
15. Given = , find BC. 16. Given = , find BC.
BC ED BC ED

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Notes: 6.3 Use Similar Polygons

____________________________________________

Similar Polygons: SYMBOL for SIMILAR: ________

Corresponding angles are _____________________________________________

Corresponding sides are _______________________________________________

Writing Similarity Statements:

Corresponding <’s:

Proportional Sides:
VA BC = = =
VXYZ

If 2 polygons are _____________, then the ratio of the lengths of 2

corresponding sides is called the ___________________.

What is the scale factor of ∆ABC to ∆XYZ? ________________

Practice:

1.) If polygon LMNO ~HIJK , completing proportions and congruence statements.

a. �M @ __?__ b. �K @ __?__ c. �N @ __?__ Hint: Draw a

diagram!!

MN ? HK HI IJ HK
d. = e. = f. =
IJ JK ? LM MN ?

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2.) In the diagram, polygon ABCD ~ GHIJ.
8 y
A B G H

5.5
11 11
x
J I
8
D C
x
a. Find the scale factor of polygon b. Find the scale factor of polygon
ABCD to polygon GHIJ. GHIJ to polygon ABCD.

c. Find the values of x and y. d. Find the perimeter of each polygon.

e. Find the ratio to the perimeter of ABCD to perimeter of GHIJ.

If 2 polygons are ___________, then the ratio of their perimeters is

equal to the ratios of their ___________.

3.) The ratio of one side of ∆ABC to the corresponding side of similar ∆DEF is 3:5.
The perimeter of ∆DEF is 48in. What is the perimeter of ∆ABC?

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Similar Triangle Postulates:
Postulate: Example: Practice with Hidden
Tools:
AA: ∆SVR and ∆UVT
If two angles of one triangle
are congruent to two angles
of another triangle, then the
two triangles are
___________________
SSS: ∆RQU and ∆SQU
If the corresponding side
lengths of two triangles are
_____________________
then the triangles are similar

SAS: Match order!!

If an angle of one triangle is
congruent to an angle of a
second triangle and the
lengths of the sides including
these angles are
_______________, then the
triangles are similar.

Ex. 1: Practice with AA: Determine if the two triangles are similar by AA.
a. ∆ABE and ∆ACD b. ∆DEC and ∆GHK c. ∆CDE and ∆ BDA

Ex. 2: Practice with SSS: Determine which triangle is similar to ∆ABC by SS. Write a similarity
statement and find the scale factor?

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Ex. 3: Practice with SAS: Determine if the two triangles are similar by SAS.

a. ∆LNM and ∆JNK b. ∆CDB and ∆CEA

Ex. 4: Mixed Practice: Determine whether the triangles are similar. If they are, state what
postulate or theorem you used and write a similarity statement.

1.) 2.)

3.)

Show that the two triangles are similar. Write a similarity statement.

4.) ∆ABE and ∆ACD 5.) ∆SVR and ∆UVT

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6.) ∆SRT and ∆PNQ 7.) ∆HGJ and ∆HFK

8.) A flagpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby
who is five feet four inches tall casts a shadow that is 40 inches long. How tall is the flagpole to
the nearest foot?

9.) Find the value of x that makes ∆ABC ~∆DEF.

KEY CONCEPT: If 2 triangles are ________________________, then they are

_________________________.

As a result, the scale factor will be _________.

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Proving Triangles Similar:

Complete the proofs.

1. Statements Reasons

2. Statements Reasons

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3. Statements Reasons

4.

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____________________________________________

Theorem 6.4: Triangle Proportionality Theorem

If a line parallel to one side of a triangle
intersects the other two sides, then it
divides the two sides proportionally.

Theorem 6.5: Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle

proportionally, then it is parallel to the
third side.

Practice Theorems 6.4-6.5:

1.) In the diagram, QS P UT , RS = 4, ST = 6, 2.) Determine whether PS P QR.
and QU = 9. What is the length of RQ ?

On your Own:
a. Find the length of YZ . b. Determine whether PS PQR .

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Theorem 6.6
If three parallel lines intersect two transversals,
then they divide the transversals proportionally.

Theorem 6.7
If a ray bisects an angle of a triangle, then it
divides the opposite side into segments whose
lengths are proportional to the lengths of the
other two sides.

Practice Theorems 6.6-6.7:

3.) Find the length of AB . 4.) Find the length of AB .

Use the diagrams to find the value of each variable.

5. 6. 7.

8.

Mixed Practice (Theorems 6.4-6.7)

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#9-13: Use the diagram to find the value of each variable.
9.) 10.)

11.) 12.)

13.)

#14-17: Determine the length of each segment.

14.) AG 15.) FC

16.) ED 17.) AE

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