4.3
What Makes Triangles Similar?
Pg. 12
Conditions for Triangle Similarity
4.3–What Makes Triangles Similar?
Conditions for Triangle Similarity
Now that you know what similar shapes
have in common, you are ready to turn to
a related question: How much information
do I need to know that two triangles are
similar? As you work through today's
lesson, remember that similar polygons
have corresponding angles that are equal
and corresponding sides that are
proportional.
4.21 – ARE THEY SIMILAR?
Erica thinks the triangles below might be
similar. However, she knows not to trust
the way figures look in a diagram, so she
asks for your help.
a. If two shapes are similar, what must be
true about their angles?
Corresponding angles must be congruent
b. What must be true about their sides?
Corresponding sides must be proportional
b. Measure the angles and sides of
Erica's Triangles and help her decide if the
triangles are similar or not.
40° 5.6cm
2cm
120° 20°
4.3cm
11.2cm
40°
4cm
120°
20°
8.6cm
Similar
4.22 – HOW MUCH IS ENOUGH?
Jessica is tired of measuring all the angles
and sides to determine if two triangles are
similar. "There must be an easier way,"
she thinks.
http://hotmath.com/util/hm_flash_movie_full.html?movie=/hotmat
h_help/gizmos/triangleSimilarity.swf
http://hotmath.com/util/hm_flash_movie_full.html?movie=/hotmath_
help/gizmos/triangleSimilarity.swf
Side-Side-Side Similarity:
If all 3 corresponding sides are proportional,
then the triangles are similar
E
B
A
C D
AB BC AC


DE EF DF
F
Angle-Angle Similarity:
If 2 corresponding angles are congruent,
then the triangles are similar
E
B
A
C D
A  D
B  E
F
4.23 – WHAT'S YOUR ANGLE?
Determine whether the triangles are similar.
If they are, explain why and write a similarity
statement.
77°
55°
AA~
ABC ~ DEF
47°
31°
AA~
ABX ~ RQX
AA~
VRS ~ URT
4.24 – ARE THEY SIMILAR?
Based on your conclusions, decide if each
pair of triangles below are similar. If they
are make a similarity statement. Then
determine if you are using Angle-Angle
similarity or Side-Side-Side similarity.
12 3

2
8
30 3
=
20 2
36 3
=
24 2
ABC
OPN
SSS~
16 4

12 3
20 5

16 4
52°
96°
AA~
d.
Yes, AA~
4.25 – FLOWCHARTS
Examine the triangles at right.
a. Are these triangles similar? Why?
Yes, AA~
b. Julio decided to use a diagram (called a
flowchart) to explain his reasoning.
Compare your explanation to Julio's
flowchart. Did Julio use the same
reasoning you used?
Yes
c. What appears to go in the bubbles of a
flowchart? What goes outside the
bubbles?
Inside:
Statements
Outside:
Reasons
4.26 – WRITING FLOWCHARTS
Besides showing your reasoning, a
flowchart can be used to organize your
work as you determine whether or not
triangles are similar.
a. Are these triangles similar? Why?
Yes,
SSS~
b. What facts must you know to use the
triangle similarity conjecture you chose?
Julio started to list the facts in a flowchart
at right. Complete the third oval.
16
4
4
given
c. Once you have the needed facts in
place, you can conclude that you have
similar triangles. Add to your flowchart by
making an oval and filling in your
conclusion.
16
4
4
CDF ~ RTQ
given
SSS ~
d. Finally, draw arrows to show the flow of
the facts that lead to your conclusion and
record the similarity conjecture you used,
following Julio's example.
16
4
4
CDF ~ RTQ
given
SSS ~
4.27 – START FROM SCRATCH
a. What is the best conjecture to test for
these triangles? SSS~ or AA~?
AA~, Only
know angles
b. Are these triangles similar? Justify
your conclusion using a flow chart.
50
L  X
K  Z
given
 sum
ΔJKL ~ ΔYZX
AA~
4.27 – HOW MANY OVALS?
a. What is the best conjecture to test for
these triangles? SSS~ or AA~?
SSS~, Only
know sides
b. Are these triangles similar? Justify
your conclusion using a flow chart.
9= 3
6 2
given
12 = 3
8 2
given
ΔABC ~ ΔDEF
SSS~
15 = 3
10 2
.
given