holt ca course 1 5-6 similar figures and proportions preparation for ns1.3 use proportions to solve...
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Holt CA Course 1
5-6 Similar Figures and Proportions
Preparation for NS1.3 Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
California Standards
4 7
N 21
Holt CA Course 1
5-6 Similar Figures and Proportions
VocabularySimilar
corresponding sides
corresponding angles
Figures with the same shape but not necessarily the same size. Two figures are similar if; the measures of their corresponding angles are equal and the ratios of the lengths of the corresponding sides are proportional.
Angles of two or more polygons that are in the same relative position.
Sides of two or more polygons that are in the same relative position.
Holt CA Course 1
5-6 Similar Figures and Proportions
Corresponding angles of two or more polygons are in the same relative position. Corresponding sides of two or more figures are between corresponding angles.
Similar figures have the same shape but not necessarily the same size. The symbol ~ means “is similar to.”
Holt CA Course 1
5-6 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.
Holt CA Course 1
5-6 Similar Figures and Proportions
Tell whether the triangles are similar.
Example 1: Determining Whether Two Triangles Are Similar
A C
B
10 in
4 in 7 in
AB corresponds to DE.BC corresponds to EF.
ABDE
=? BC
EF=? AC
DF4
167
281040
14
14
14
Since the measures 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.
=? =?
The corresponding angles of the figures have equal measures.
D
E
F
16 in 28 in
40 in
Holt CA Course 1
5-6 Similar Figures and Proportions
When naming similar figures, list the letters of the corresponding angles in the same order.
Writing Math
Holt CA Course 1
5-6 Similar Figures and ProportionsFor triangles, if the corresponding side lengths are all proportional, then the corresponding angles must have equal measures. For figures that have four or more sides, if the corresponding side lengths are all proportional, then the corresponding angles may or may not have equal measures.
Holt CA Course 1
5-6 Similar Figures and Proportions
Tell whether the figures are similar.
Example 2: Determining Whether Two Four-Sided Figures are Similar
The corresponding angles of the figures have equal measure.
Write each set of corresponding sides as a ratio.
Holt CA Course 1
5-6 Similar Figures and Proportions
Example 2 Continued
MNQR MN corresponds to QR.
NORS
OPST
MPQT
NO corresponds to RS.
OP corresponds to ST.
MP corresponds to QT.
Holt CA Course 1
5-6 Similar Figures and Proportions
Determine whether the ratios of the lengths of the corresponding sides are proportional.
Example 2 Continued
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=
? 812 =
? 4 6 =
? 1015
23
= 23
= 23
= 23
Holt CA Course 1
5-6 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
Holt CA Course 1
5-6 Similar Figures and Proportions
Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.
Check It Out! Example 1
A C
B
9 in
3 in 7 in
AB corresponds to DE.BC corresponds to EF.
ABDE
=? BCEF
=? ACDF
39
721
927
13
13
13
Since the measures 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.
=? =?
The corresponding angles of the figures have equal measures.
D
E
F
9 in 21 in
27 in
Holt CA Course 1
5-6 Similar Figures and Proportions
Tell whether the figures are similar.
Check It Out! Example 2
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.
Holt CA Course 1
5-6 Similar Figures and Proportions
Check It Out! Example 2 Continued
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
Holt CA Course 1
5-6 Similar Figures and Proportions
Determine whether the ratios of the lengths of the corresponding sides are proportional.
Check It Out! Example 2 Continued
Write ratios using correspondingsides.
Substitute the length of the sides.
Simplify each ratio.
MNQR
=? NORS
=? OPST
=? MPQT
60240 =?
80320 =
? 47.5190 =
? 100400
14
= 14
= 14
= 14
Since the measure of the corresponding angles are equal and 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