Similar and Congruent Polygons - Warrick County … · Lesson 10.5 Similar and Congruent Polygons...
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502 Chapter 10 Geometric Figures Two polygons are if they have the same shape but not necessarily the same size. The symbol S is used to indicate that two polygons are similar. are similar polygons that have the same shape and the same size. Congruent polygons similar Word Watch similar polygons, p. 502 congruent polygons, p. 502 When naming congruent or similar polygons, list the letters for the corresponding vertices in the same order. For instance, in Example 1 you cannot write “TRST cTYXZ” because aR and aY are not corresponding angles. W a t c h O u t ! BEFORE Now WHY? Similar and Congruent Polygons You classified polygons. You’ll use properties of similar and congruent polygons. So you can compare two sails, as in Exs. 2–3. EXAMPLE 1 Finding Measures of Congruent Polygons Given that T RST cT XYZ, name the corresponding sides and corresponding angles. Then find XY. Corresponding sides: RS & and XY &*, RT &* and XZ &*, ST &* and YZ &* Corresponding angles: aR and a X, aS and aY, aT and a Z Because XY &* and RS & are corresponding sides, they are equal in length. XY RS 6 inches Y Z X 11 in. 6 in. S R T 8 in. with Reading If two polygons are similar, then the ratio of the lengths of corresponding sides is called the scale factor. T LMN ST PQR T ABC c T DEF Similar Polygons Congruent Polygons B A C 10 m 6 m 8 m R P Q 5 m 3 m 4 m N L M F D E Angles Corresponding angles are congruent: aA caD, aB caE, and aC caF Sides Corresponding sides are congruent: AB &c DE &, AC &c DF &, and BC &c EF & Angles Corresponding angles are congruent: aL caP, aM caQ, and aN caR Sides Ratios of lengths of corresponding sides are equal: L P M Q M QR N P LN R
Transcript of Similar and Congruent Polygons - Warrick County … · Lesson 10.5 Similar and Congruent Polygons...
502 Chapter 10 Geometric Figures
Two polygons are if they have the same shape but notnecessarily the same size. The symbol S is used to indicate that twopolygons are similar. are similar polygons thathave the same shape and the same size.
Congruent polygons
similar
Word Watchsimilar polygons, p. 502congruent polygons, p. 502
When namingcongruent or similarpolygons, list the letters forthe corresponding verticesin the same order. Forinstance, in Example 1 you cannot write “TRST c TYXZ” becauseaR and aY are notcorresponding angles.
Watch Out!
B E F O R E Now W H Y ?
Similar and Congruent Polygons
You classified polygons. You’ll use properties of similarand congruent polygons.
So you can compare two sails,as in Exs. 2–3.
E X A M P L E 1 Finding Measures of Congruent Polygons
Given that TRST c TXYZ,name the corresponding sides and corresponding angles. Then find XY.
Corresponding sides: RS& and XY&*, RT&* and XZ&*, ST&* and YZ&*
Corresponding angles: aR and aX, aS and aY, aT and aZ
Because XY&* and RS& are corresponding sides, they are equal in length.
XY � RS � 6 inches
Y
Z
X
11 in.6 in.
S
R
T8 in.
with Reading
If two polygons are similar,then the ratio of thelengths of correspondingsides is called the scalefactor.
TLMN S TPQR TABC c TDEF
Similar Polygons Congruent Polygons
BA
C
10 m6 m
8 m RP
Q
5 m3 m
4 m NL
M
F
DE
Angles Corresponding angles arecongruent:aA c aD,aB c aE, and aC c aF
Sides Corresponding sides arecongruent:
AB& c DE&, AC& c DF&, and BC& c EF&
Angles Corresponding angles arecongruent:aL c aP,aM c aQ, and aN c aR
Sides Ratios of lengths of correspondingsides are equal:
�LPMQ� � �
MQR
N� � �
PLNR�
Use the fact that ABCD c EFGH.
1. Name the corresponding sides and corresponding angles.
2. Find the unknown angle measures.
Your turn now
E X A M P L E 2 Finding the Ratio of Lengths
Given that TABC S TDEF,find the ratio of the lengths of the corresponding sidesof TABC to TDEF.
Write a ratio for each pair of corresponding sides. Then substitute thelengths of the sides and simplify each ratio.
�DAB
E� � �
11
62� � �
43
� �BE
CF� � �
21
05� � �
43
� �DAC
F� � �
22
81� � �
43
�
ANSWER The ratio of the lengths of the corresponding sides is �43
�.
E X A M P L E 3 Checking for Similarity
Painting A rectangular painting has a length of 48 inches and a width of36 inches. A rectangular print of the painting has a length of 32 inchesand a width of 24 inches. Are the original and the print similar figures?
Solution
Because both figures are rectangles, all angles are right angles, socorresponding angles are congruent. To determine whether the figuresare similar, see if the ratios of the lengths of the corresponding sides areequal.
��W
Wid
itdhth
ofo
of
rpirgiinn
tal
�
�43
82� � �
32
64� Substitute values.
�32
� � �32
� Simplify.
ANSWER The corresponding angles are congruent and the ratios of thelengths of the corresponding sides are equal, so the figures are similar.
Write ratios for lengths ofcorresponding sides.
Length of original���
Length of print
B
A
E H
FDG
C135�
60� 75�?
? ?
?
16 cm 12 cm
28 cm21 cm
20 cm 15 cm
CF
B E
AD
Lesson 10.5 Similar and Congruent Polygons 503
with Solving
The ratios �DAB
E�, �
BECF�, and �
ADCF�
are the ratios for thelengths of thecorresponding sides ofTDEF to TABC inExample 2. Using theseratios, the ratio of thelengths of the
corresponding sides is �34
�.
■ PaintingAlexandra Nechita’s first artexhibit was held when shewas 8 years old. Her painting“My Future Rainbow” is a 36 in. by 24 in. rectangle.Her painting “SharingMoments” is a 28 in. by 22 in. rectangle. Are the twopaintings similar figures?
Art
ExercisesMore Practice, p. 714
504 Chapter 10 Geometric Figures
Name the corresponding sides and the corresponding angles of thecongruent polygons. Then find the unknown measures.
4. KLMNc QRST 5. TFGH cTJKL
Tell whether the two polygons are similar. If they are similar, find theratio of the lengths of the corresponding sides of figure A to figure B.
6. 7.
8. 9.
J L
K
35 cm
30 cm
40 cm
R P
Q
32 cm
24 cm
28 cmAB
U W
V
16 in.
14 in.
19 in.
R T
S
11 in.
9 in.
12 in.A
B
H G
EF
6 m
10 m
12 mD C
BA3 m
4 m5 m
6 m
2 m A
B
X Y
W Z
6 yd
8 yd E F
GD
3 yd
4 yd
AB
8 yd
6 yd4 yd
3 yd
J
L K
F
G H
?
75� 60� ??
45�
Q
T S
R
?
?
3 m
?
K
N M
L
4 m
?
?
?
Practice and Problem Solving
1. Vocabulary Copy and complete: Two polygons that have the sameshape but not necessarily the same size are _?_ polygons.
Sails Use the diagram of the two sailboat sails and the fact thatTABC S TDEF.
2. Name the corresponding sides and thecorresponding angles.
3. Find the ratio of the lengths of thecorresponding sides of TABC to TDEF.
Getting Ready to Practice
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Example Exercises1 4–52 6–93 6–10
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16 ft24 ft
6 ft
14 ft21 ft
9 ft
B
A C
E
D F
Lesson 10.5 Similar and Congruent Polygons 505
10. Photo Sticker A standard photograph is a rectangle with a length of 6 inches and a width of 4 inches. You have a camera that makes photostickers. Each sticker is a rectangle with a length of 1.4 inches and awidth of 0.9 inch. Are the photograph and sticker similar figures?Explain your reasoning.
Critical Thinking Tell whether the statement is true or false. If it isfalse, give an example to show why.
11. Two rectangles are congruent if they have the same perimeter.
12. Two squares are congruent if they have the same perimeter.
13. Critical Thinking Draw a rectanglesimilar to rectangle DEFG shown.Then draw a rectangle congruent toDEFG. Explain why each rectangle issimilar or congruent to DEFG.
14. Athletic Fields An NCAA football field has a width of 160 feet and alength of 360 feet. An NCAA soccer field can vary from 195 feet to 240 feet in width and from 330 feet to 360 feet in length. Can NCAAfootball and soccer fields ever be similar rectangles? Explain.
15. Challenge Two rectangles are similar. The ratio of the lengths of theircorresponding sides is 1 : 2. Find the ratio of the perimeters of the tworectangles. Then find the ratio of the areas. Explain your answers.
16. Use equivalent ratios to solve the proportion �11
58� � �
6x
�. (Lesson 8.4)
17. Find the perimeter of a regular quadrilateral with a side length of 16 meters. (Lesson 10.4)
Test the number for divisibility by 2, 3, 5, 6, 9, and 10.
18. 136 19. 942 20. 1675 21. 2458
Basic Skills
Mixed Review
E F
D G
4 in.
5 in.
22. Multiple Choice If TABCcTDEF and both triangles are scalene,which of the following is not true?
A. ACwc DFwx B. ABwc EFw C. aBc aE D. aC c aF
23. Short Response Rectangle JKLM has a length of 30 feet and a width of9 feet. Rectangle NPQR has a length of 50 feet and a width of 15 feet.Tell whether the two rectangles are similar. Explain your reasoning.
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