Similarity

1
Similarity

• Unit 7

2
Similar triangles
Similar polygons
Similarity
Parallel
Parts of similar triangles
Lines
3
Similar Polygons

• Polygons in which
• Corresponding angles are congruent
• Corresponding sides are proportional

4
Scale factor

• Ratio of the lengths of the corresponding sides
  of 2 similar polygons
• Scale factor 15/25 3/5

25
15
5
Missing sides

• find missing sides using proportions
• 15 x
• 25 20

25x 2015 x 300/25 x 12
25
20
15
x
6
Similar triangles

• AA Similarity
• SSS Similarity
• SAS Similarity

7
AA Similarity (angle-angle)

• If 2 angles of one triangle are congruent to 2
  angles of another triangle, then the triangles
  are similar

8
AA Similarity

B
C
?
A
E
If ? A ? ? D and ? B ? ?E
F
D
9
SSS Similarity (side-side-side)

• If the measures of the corresponding sides of
  two triangles are proportional, then the
  triangles are similar

10
SSS Similarity

B
E
?
C
A
If AB BC AC DE EF DF
D
F
11
SAS Similarity (side-angle-side)

• If the measures of two sides of a triangle are
  proportional to the measures of the corresponding
  sides of another triangle and the included
  angles are congruent, then the triangles are
  similar

12
SAS Similarity

B
E
?
A
C
If AB BC and ? B ? ?E DE EF
D
F
13
Theorem 7-3

• Similarity of triangles is
• reflexive
• symmetric
• transitive

14
Triangle proportionality
If a line is parallel to one side of a triangle
and intersects the other 2 sides in distinct
points, then it separates those sides into
segments of proportional lengths.
Theorem 7-4
15
Triangle proportionality
If BD AE, then AB DE AB BC BC
DC DE DC
C
D
B
A
E
16
Converse of Triangle proportionality
If a line intersects 2 sides of a triangle and
separates the sides into segments of proportional
lengths, then the line is parallel to the 3rd
side of the triangle.
Theorem 7-5
17
Converse of Triangle proportionality
If AB DE AB BC BC DC DE DC then
BD AE
C
D
B
A
E
18
Theorem 7-6
A segment whose endpoints are the midpoints of
2 sides of a triangle is parallel to the third
side and its length is ½ the length of the 3rd
side.
19
Theorem 7-6
If B and D are midpoints, then BD AE and BD
½ AE
C
D
B
A
E
20
Corollary 7-1
If 3 or more parallel lines intersect 2
transversals, then they cut off the transversals
proportionally.
21
AB DE BC EF
A
D
E
B
C
F
22
Corollary 7-2
If 3 or more parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal.
23
AB BC then DE EF
If AD BE, BE CF,
A
D
E
B
C
F
24
Theorem 7-7
If 2 triangles are similar, then the perimeters
are proportional to the measures of the
corresponding sides.
25
Theorem 7-8
If 2 triangles are similar, then the measures
of the altitudes are proportional to the measures
of the corresponding sides.
26
Theorem 7-9
If 2 triangles are similar, then the measures
of the angle bisectors are proportional to the
measures of the corresponding sides.
27
Theorem 7-10
If 2 triangles are similar, then the measures
of the medians are proportional to the measures
of the corresponding sides.
28
Theorem 7-11
An angle bisector in a triangle separates the
opposite side into segments that have the same
ratio as the other 2 sides.