Download Triangle Congruence Shortcuts Investigation Packet

Ch 4 Triangle Congruence Shortcut Investigation Key
Name __________________________
Triangle Congruence Shortcuts Investigation Packet
The Big Question: How many parts of a triangle do you need to duplicate in order to guarantee that you
have congruent triangles?
According to the definition of congruent triangles, you would need to know that all six pairs of
corresponding parts were congruent. For example, to show that
ΔABC ≅ ΔTRI , you would have to show all of the following
B
corresponding parts congruent :
They have the same shape,
the corresponding angles are
congruent.
∠A ≅ ∠T
∠B ≅ ∠R
∠C ≅ ∠I
They have the same size,
the corresponding sides are
congruent.
AB ≅ TR
BC ≅ RI
AC ≅ TI
C
A
R
I
T
The purpose of this investigation is to see if you can get (or duplicate) congruent triangles with less than 6
parts? What is the minimum number of parts that you would need to duplicate in order to create
congruent triangles?
Duplicating only one part surely won’t create congruent triangles!
How about two parts? (Note: S = side and A = angle)
Below, show that by copying two sides, two angles, or
a side and an angle will probably not give congruent
triangles. (Goal is to make two different
triangles…show that they are not congruent!)
SS Congruence?
AA Congruence?
x
A
T
SA Congruence?
x
y
x
y
Not Congruent
S. Stirling
Page 1 of 5
y
Ch 4 Triangle Congruence Shortcut Investigation Key
Name __________________________
Lesson 4.4 Are There Congruence Shortcuts? SSS, SAS, and SSA
Is there any way to make congruent triangles (duplicate triangles) with 3 parts?
Three Parts (Part 1: at least two pairs of sides equal. )
On all of the investigations, use the given method to try to draw or construct a triangle congruent to the given
triangle, ΔABC ≅ ΔXYZ . Try to get two non-congruent triangles! Can you do it? If the triangles have the
same size and shape, they are congruent. If you can create two different triangles from the given parts, then that
method does not guarantee congruence.
SSS Congruence Conjecture Does SSS guarantee congruent triangles? YES
If the three sides of one triangle are congruent to the three sides of another triangle, then
the triangles are congruent.
B
Y
A
Z
X
C
SAS Congruence Conjecture Does SAS guarantee congruent triangles? YES
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the triangles are congruent.
B
Y
A
C
Z
X
SSA or ASS Congruence Does SSA guarantee congruent triangles? NO
If two sides and the non-included angle of one triangle are congruent to two sides and the
non-included angle of another triangle, then the triangles are NOT necessarily congruent.
Y
B
A
S. Stirling
C
X
X
Z
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Ch 4 Triangle Congruence Shortcut Investigation Key
Name __________________________
Lesson 4.5 ASA, SAA, and AAA Congruence Shortcuts?
Three Parts (Part 2: at least two pairs of angles equal.)
On all of the investigations, use the given method to try to draw or construct a triangle congruent to the given
triangle, ΔABC ≅ ΔXYZ . Try to get two non-congruent triangles! Can you do it? If the triangles have the
same size and shape, they are congruent. If you can create two different triangles from the given parts, then that
method does not guarantee congruence.
ASA Congruence Conjecture Does ASA guarantee congruent triangles? YES
If two angles and the included side of one triangle are congruent to two angles and the
included side of another triangle, then the triangles are congruent.
B
Y
A
C
Z
X
SAA or AAS Congruence Conjecture Does SAA guarantee congruent triangles? YES
If two angles and a non-included side of one triangle are congruent to the corresponding
angles and side of another triangle, then the triangles are congruent.
Hint: Find the measure of the third angle first. Then do ASA.
B
Y
A
C
Z
X
AAA Congruence Conjecture Does AAA guarantee congruent triangles? NO
If three angles of one triangle are congruent to the corresponding angles of another
triangle, then the triangles are NOT necessarily congruent.
B
Y
A
S. Stirling
C
X
Z
Page 3 of 5
Ch 4 Triangle Congruence Shortcut Investigation Key
Name __________________________
Complete the Ch 4 Note Sheet, page 6 – 9.
How do you apply the congruence short cuts?
Steps to determining congruence:
1. Make sure corresponding vertices match up.
2. Do you have congruence (SSS, SAS, ASA or AAS)? Make sure corresponding parts match up!
3. If not, find any equal parts (sides or angles) using conjectures you already know. Mark you diagram!
4. Repeat steps 2 and 3 until you get congruence or decide that congruence “cannot be determined”.
In Exercises 1–3, name the conjecture that leads to each congruence.
ASA or AAS Cong.
SSS Cong.
SSS Cong.
In Exercises 4–8, name a triangle congruent to the given triangle and state the congruence conjecture. If
you cannot show any triangles to be congruent from the information given, write “cannot be determined”
and redraw the triangles so that they are clearly not congruent.
ΔAPM ≅ ΔBQM
ΔKIE ≅ ΔTIE
ΔABC ≅ ΔXYZ
SAS Cong.
SSS Cong.
AAS Cong.
ΔMON ≅ ΔTNO
ΔSQR ≅ ΔUTR
SAS Cong.
SSA NOT Cong.
Cannot be determined!
S. Stirling
Page 4 of 5
Ch 4 Triangle Congruence Shortcut Investigation Key
Name __________________________
MORE Examples: How do you apply the congruence short cuts?
In Exercises 1–6, name a triangle congruent to the given triangle and state the congruence conjecture. If
you cannot show any triangles to be congruent from the information given, write “cannot be determined”
and explain why.
ΔPIT ≅ ΔTOP
ΔXVW ≅ ΔXZY
ΔECD ≅ ΔACB
SSA NOT Cong.
Cannot be determined!
ASA or AAS Cong.
ASA or AAS Cong.
ΔPQS ≅ ΔPRS
ΔACN ≅ ΔNRA
ΔEQL ≅ ΔGQK
ASA Cong.
Cannot be determined!
AAS or ASA Cong.
AAS or ASA Cong.
Match sides: 125 = x + 55, x = 70
350 = x + x + 55 + 2x + 15
If x = 70, it works.
So cong. by SSS.
S. Stirling
95 = x + 25 + 2x – 10 + x
95 = 15 + 4x so x = 4
Match sides: Is TV = VW? No
4 + 25 ≠ 40!
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