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Similar Figures and Proportions
Warm UpWarm UpProblem of the DayProblem of the DayLesson PresentationLesson Presentation
Warm UpFind the cross products, then tell whether the ratios are equal.
Course 2
5-7 Similar Figures and Proportions
1. 166
, 4015
2. 38
, 1846
3. 89
, 2427
4. 2812
, 4218
240 = 240; equal
216 = 216; equal
504 = 504; equal
138 = 144; not equal
Problem of the DayEvery 8th telephone pole along a road has a red band painted on it. Every 14th pole has an emergency call phone on it. What is the number of the first pole with both a red band and a call phone? 56
Course 2
5-7 Similar Figures and Proportions
Learn to use ratios to determine if two figures are similar.
Course 2
5-7 Similar Figures and Proportions
Vocabularysimilarcorresponding sidescorresponding angles
Insert Lesson Title Here
Course 2
5-7 Similar Figures and Proportions
Course 2
5-7 Similar Figures and Proportions
Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similar figures have the same shape but not necessarily the same size.
Course 2
5-7 Similar Figures and Proportions
When naming similar figures, list the letters of the corresponding vertices in the same order. In the previous table ∆ABC ~ ∆DEF.
Writing Math
Course 2
5-7 Similar Figures and Proportions
Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles.
82◦
Corresponding angles
D
E
F
Corresponding sides
A
B
C
82◦
55◦43◦ 55◦43◦
Course 2
5-7 Similar Figures and Proportions
SIMILAR FIGURESTwo figures are similar if• The measures of their corresponding angles are
equal. • The ratios of the lengths of the corresponding
sides are proportional.
Course 2
5-7 Similar Figures and Proportions
A side of a figure can be named by its endpoints, with a bar above.
AB
Without the bar, the letters indicate the length of the side.
Reading Math
Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.
Additional Example 1: Determining Whether Two Triangles Are Similar
Course 2
5-7 Similar Figures and Proportions
A C
B
10 in4 in 7 in D
E
F
16 in 28 in
40 in
AB corresponds to DE.BC corresponds to EF.
ABDE =? BC
EF =? ACDF
416
728
1040
14
14
14
Since the ratios of the corresponding sides are equivalent, the triangles are similar.
Write ratios using the corresponding sides.
Substitute the length of the sides.
Simplify each ratio.
=? =?
AC corresponds to DF.
=? =?
Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.
Check It Out: Example 1
Course 2
5-7 Similar Figures and Proportions
A C
B
9 in3 in 7 in D
E
F
9 in 21 in
27 in
AB corresponds to DE.BC corresponds to EF.
ABDE =? BC
EF =? ACDF
39
721
927
13
13
13
Since the ratios of the corresponding sides are equivalent, the triangles are similar.
Write ratios using the corresponding sides.
Substitute the length of the sides.
Simplify each ratio.
=? =?
AC corresponds to DF.
=? =?
Tell whether the figures are similar.
Additional Example 2: Determining Whether Two Four-Sided Figures are Similar
Course 2
5-7 Similar Figures and Proportions
The corresponding angles of the figures have equal measure.
Write each set of corresponding sides as a ratio.
Additional Example 2 Continued
Course 2
5-7 Similar Figures and Proportions
MNQR MN corresponds to QR.
NORS
OPST
MPQT
NO corresponds to RS.
OP corresponds to ST.
MP corresponds to QT.
Determine whether the ratios of the lengths of the corresponding sides are proportional.
Additional Example 2 Continued
Course 2
5-7 Similar Figures and Proportions
Write ratios using corresponding sides.
Substitute the length of the sides.
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the figures are similar.
MNQR
=? NORS
=? OPST
=? MPQT
6 9=? 8
12 =? 4 6 =? 10
15
23 = 2
3 = 23 = 2
3? ? ?
Tell whether the figures are similar.
Check It Out: Example 2
Course 2
5-7 Similar Figures and Proportions
The corresponding angles of the figures have equal measure.
100 m
80 m
60 m 47.5 m80°
90° 125°65°
M P
NO
400 m
320 m
190 m240 m
80° 65°
90° 125°
Q T
R S
Write each set of corresponding sides as a ratio.
Check It Out: Example 2 Continued
Course 2
5-7 Similar Figures and Proportions
MNQR MN corresponds to QR.
NORS
OPST
MPQT
NO corresponds to RS.
OP corresponds to ST.
MP corresponds to QT.
100 m
80 m
60 m 47.5 m80°
90° 125°65°
M P
NO
400 m
320 m
190 m240 m
80° 65°
90° 125°
Q T
R S
Determine whether the ratios of the lengths of the corresponding sides are proportional.
Check It Out: Example 2 Continued
Course 2
5-7 Similar Figures and Proportions
Write ratios using correspondingsides.
Substitute the length of the sides.
Simplify each ratio.
Since the ratios of the corresponding sides are equivalent, the figures are similar.
100 m
80 m
60 m 47.5 m80°
90° 125°65°
M P
NO
400 m
320 m
190 m240 m
80° 65°
90° 125°
Q T
R S
MNQR
=? NORS
=? OPST
=? MPQT
60240 =? 80
320 =? 47.5190 = ? 100
400
14 = 1
4 = 14 = 1
4? ? ?
Lesson Quiz: Part I
Insert Lesson Title Here
Course 2
5-7 Similar Figures and Proportions
Tell whether the figures are similar.1. similar59°
59°
35°35°
86°86°