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Study Guide for Chapter 7
7.1 Ratios, Proportions
Vocabulary: ratio, extended rations, proportion, cross product
Example 1
Simplify ratios (reduce ratios)
a. 76 cm: 8 cm
b. 4 ft
24in.
Example 2 Use a ratio to find a dimension
a. Painting You are painting barn doors. You know that the perimeter of the doors
is 64 feet and that the ratio of the length to the height is 3:5. Find the area of the
doors. (hint. create an equation: 3x + 5x = 64 to find the width and length)
b. The perimeter of a rectangular table is 21 feet and the ratio of its length to its
width is 5:2. Find the length and width of the table. (hint. use 2W + 2L = P,
where P is the perimeter, W the width, L the length).
Example 3 Use extended ratios
The measures of the angles in BCD are in the extended ratio of 2:3:4. Find the
measures of the angles. (hint. create an equation, 2x + 3x + 4x = 180)
How to set up proportions:
1. Cross Products Property In a proportion, the product of the extremes equals the
a c
product of the means. If = where b  0 and d  0, then ad = bc
b d
The following proportions are equivalent, where a, b, c, d represent numbers:
a c
=
b d
b d
=
a c
a b
=
c d
c d
=
a b
Solving a real-world problem:
Bowling. You want to find the total number of rows of boards that make up 24
lanes at a bowling alley. You know that there are 117 rows in 3 lanes. Find the
total number of rows of boards that make up the 24 lanes.
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7.2 Similar Polygons (pg. 469)
CORRESPONDING LENGTHS IN SIMILAR POLYGONS
If two polygons are similar, then the ratio of any two corresponding lengths in the
polygons is equal to the ____________ of the similar polygons.
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their
corresponding side lengths.
If KLMN  PQRS, then KL  LM  MN  NK = ____ = ____ = ____ = ____.
PQ  QR  RS  SP
Example 1
Use properties of proportions for triangles
In the diagram,
AC
DF

BC . Write four true proportions.
EF
=====================================================
7.3 Similar Triangles (pg. 478)
Know how to use proportions to identify similar polygons and use similarity
statements
Example 1 In the diagram, ∆ ABC  ∆ DEF.
a. List all pairs of congruent angles.
b. Check that the ratios of corresponding side lengths are equal.
c. Write the ratios of the corresponding side lengths in statement of proportionality.
Example 2 If the triangles to the right are
similar, then find x. What is the scale factor?
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More Practice: In the diagrams, ∆PQR  ∆WXY.
1. Find the perimeter of ∆WXY.
Example 4
Find perimeters of similar figures
Basketball A larger cement court is being poured for a basketball hoop in place of a
smaller one. The court will be 20 feet wide and 25 feet long. The old court was similar in
shape, but only 16 feet wide.
a. Find the scale factor of the new court to the old court.
Find the perimeters of the new court and the old court.
Prove Triangles Similar by AA
Be able to use the AA Similarity Postulate.
POSTULATE 7.1: ANGLE-ANGLE (AA) SIMILARITY POSTULATE
If two angles of one triangle are congruent to two angles of another triangle, then the two
triangles are similar.
∆JKL ~ ∆XYZ
Example 1 Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are, write a similarity
statement. Explain your reasoning.
Example 2 Show that the two triangles are similar.
a. ∆RTV and ∆RQS
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Prove Triangles Similar by SSS and SAS
Be able to use the SSS and SAS Similarity Theorems.
THEOREM 7.2: SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM
If the corresponding side lengths of two triangles are _____________, then the triangles
are similar.
If
=
=
, then ABC ~ RST.
Example 1 Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
Example 2
Use the SSS Similarity Theorem
Find the value of x that makes ABC ~ DEF.
THEOREM 7.3: SIDE-ANGLE-SIDE (SAS) SIMILARITY THEOREM
If the lengths of 2 sides of one triangle are _____________________ to the lengths of 2
corresponding sides of another ____________________ and the included angles are
congruent, then the triangles are similar.
If X  M , and
Z
X
=
, then XYZ  MNP.
PM MN
TRIANGLE SIMILARITY POSTULATE AND THEOREMS
AA Similarity Postulate If A  D and B  E, then ABC ~ DEF.
SSS Similarity Theorem If
AB
BC
AC
=
=
, then ABC ~ DEF.
DE EF
DF
AB
AC
DE DF
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SAS Similarity Theorem If A  D and
=
, then ABC  DEF.
Example 4 Choose a method
Tell what method you would use to show that the triangles are similar.
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7.4 Parallel Lines and Proportional Parts (Pg. 490)
Be able to use proportions with a triangle parts or parts of transversals with parallel lines.
THEOREM 7.5: TRIANGLE PROPORTIONALITY THEOREM
If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides
into __________________ of _____________________ lengths.
If TU | QS , then
=
_____
.
_____
Theorem 7.6: CONVERSE OF THE TRIANGLE PROPORTIONALITY THM
If a line divides two sides of a triangle proportionally, then it is parallel to the _____________side of the
triangle .
If
RT
TQ

RU
US
, then
||
.
_____ _____
Example 1 Find the length of a segment
In the diagram,
QS | UT , RQ = 10, RS = 12, and ST = 6. What is the length of
QU ?
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Corollary 7.1
If three parallel lines intersect two transversals, then they divide the transversals ______________ .
UW
WY
Know also corollary 7.2, pg. 493
7-5 Parts of Similar Triangles, or Special Segments of Similar Triangles
7.8 If two triangles are similar, the lengths of corresponding altitudes are
proportional to the lengths of corresponding sides.
7.9 If 2 triangles are similar, the lengths of corresponding angle bisectors are
proportional to the lengths of corresponding sides.
7.10 If two triangles are similar, the lengths of corresponding medians are
proportional to the lengths of corresponding sides.
7.11 If a ray bisects an angle of a triangle, then it divides the opposite side into
segments whose lengths are _____________ to the lengths of the other two sides.
AD
DB
Example 3 Use Corollary 7.1
Farming A farmer’s land is divided by a newly constructed interstate. The distances shown are in meters.
Find the distance CA between the north border and the south border of the farmer’s land.
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Use Theorem 6.6.
Example 4 Use Theorem 7.9
In the diagram, DEG  GEF. Use the given side lengths to find the length of
DG .
EG
7-6 Similarity Transformations (Pg. 511)
VOCABULARY –
What is a similarity transformation?
What is a dilation?
What is the scale factor for a dilation
Name 2 types of dilations
An enlargement is when the scale factor is > _____________.
A reduction is when the scale factor is < _________.
Practice problems 1-5 (all) page 514.
Practice doing more problems from page 524: 1-8 (all), 9-34 (all). These are good review
problems and will be assigned in class.
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