Are There Congruence Shortcuts?
Objectives:
Explore shortcut methods for determining whether triangles are congruent
Discover the SSS and SAS are valid congruence shortcuts but SSA is not
Homework: 4.4 pg.224-225 # 4-9, 12-14, 23, 24
Triangle Inequalities
Do now:
Quiz 4.1- 4.3
A building contractor has just assembled two massive
triangular trusses to support the roof of a recreation
hall. Before the crane hoists them into place, the
contractor needs to verify that the two triangular
trusses are identical. Must the contractor measure
and compare all six parts of both triangles?
Wikipedia:
In architecture a truss is
a structure comprising one
or more triangular units
constructed with straight
members whose ends are
connected at joints referred
to as nodes.
http://www.youtube.com/watch?v=8hmPcxOBQA
for SSA:
http://www.mathopenref.com/congruentssa.ht
ml
Explain what the picture statement means.
Create a picture statement to represent
the SAS Triangle Congruence Conjecture.
Explain what the picture statement means.
In the third investigation you discovered
that the SSA case is not a triangle
congruence shortcut. Sketch a
counterexample to show why.
Which conjecture tells you that triangles are
congruent?
Y is a midpoint
Closing the Lesson: The main points
of this lesson are that SSS and SAS
can be used to establish the
congruence of triangles but SSA
cannot.
What is the reason why SSA fails?
4.4-4.5 Are There Congruence Shortcuts?
Objectives:
Explore shortcut methods for determining whether
triangles are congruent
Discover valid congruence shortcuts
Homework: 4.5 pg.229 # 4, 6, 8, 10, 12, 14
Do Now:
# 4-6,10(!) pg. 224
Do you need all six ?
NO !
SSS
SAS
ASA
AAS
Side-Side-Side (SSS)
1. AB  DE
2. BC  EF
3. AC  DF
ABC   DEF
Side-Angle-Side (SAS)
1. AB  DE
2. A   D
3. AC  DF
ABC   DEF
included
angle
Included Angle
The angle between two sides
G
I
H
Included Angle
Name the included angle:
E
Y
S
YE and ES
E
ES and YS
S
YS and YE
Y
Angle-Side-Angle (ASA)
1. A   D
2. AB  DE
ABC   DEF
3.  B   E
included
side
Included Side
The side between two angles
GI
HI
GH
Included Side
Name the included side:
E
Y
S
Y and E
YE
E and S
ES
S and Y
SY
Angle-Angle-Side (AAS)
1. A   D
2.  B   E
ABC   DEF
3. BC  EF
Non-included
side
Warning: No SSA Conjecture
There is no such
thing as an SSA
conjecture!
E
B
F
A
C
D
NOT CONGRUENT
Warning: No AAA Conjecture
There is no such
thing as an AAA
conjecture!
E
B
A
C
D
NOT CONGRUENT
F
The Congruence Conjectures
 SSS
correspondence
 ASA
correspondence
 SAS
correspondence
 AAS
correspondence
 SSA correspondence
 AAA
correspondence
Name That Conjecture
(when possible)
SAS
SSA
ASA
SSS
Name That Conjecture
(when possible)
AAA
ASA
SAS
SSA
Name That Conjecture
(when possible)
Vertical Angle
Reflexive
Property
SAS
SAS
Vertical Angles
SAS
Reflexive
Property
SSA
Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
Name That Conjecture
(when possible)
10. The perimeter of ABC is 180 m.
Is
ABC
ADE? Which conjecture supports
your conclusion?
Name That Conjecture
(when possible)
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
For AAS:
Determine whether the triangles are congruent, and
name the congruence shortcut. If the triangles cannot
be shown to be congruent, write
“cannot be determined.”
Solve: pg.224-226 # 11-19
pg.229 # 6-9
Pg.230 # 13, 15, 18
Shortcut for showing triangle congruence allows us to avoid proving the
congruence of all six pairs of corresponding parts. There are four shortcuts for
proving triangle congruence. Students discovered SSS and SAS in the previous
lesson. Now they have discovered ASA and SAA. SSA and AAA are not
congruence shortcuts. Other arrangements of the letters, such as ASS (not a
shortcut) and AAS (a shortcut), are included among these six.
4.6 Corresponding Parts of Congruent Triangles
Objectives:
Show that pairs of angles or pairs of sides are congruent
by identifying related triangles
Homework: lesson 4.6 pg.233 # 3, 4, 5, pg. 231 # 26
Do now:
CPCTC-Corresponding Parts of Congruent
Triangles are Congruent
Paragraph proof
Together: # 1, 2 pg 233 # 12
4.6 Corresponding Parts of Congruent Triangles
Objectives:
Show that pairs of angles or pairs of sides are congruent
by identifying related triangles
Homework: lesson 4.6 pg.233-234 # 6, 7, 9, 18
CPCTC-Corresponding Parts of Congruent
Triangles are Congruent
Do now:
In Chapter 3, you used inductive reasoning to discover how to duplicate an angle
using a compass and straightedge. Now you have the skills to explain why the
construction works using deductive reasoning. The construction is shown at
right.Write a paragraph proof explaining why it works.
Practice:
# 3 (from h/w)
# 3 (from h/w)
#5 (h/w)
#6
Closure: To show that two segments or angles (the targets) are congruent, you
will often find two congruent triangles in which these segments or angles are
corresponding parts.