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,
so the corresponding angles
are congruent.
They have the same size,
so the corresponding sides are
congruent.
∠A ≅ ∠T
∠B ≅ ∠R
∠C ≅ ∠I
C
A
R
AB ≅ TR
BC ≅ RI
AC ≅ TI
T
I
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)
Is copying two sides, two angles, or a side and an angle be enough to guarantee two triangles are
congruent? Use tracing paper to confirm that the parts are congruent.
SS Congruence?
x
AA Congruence?
A
y
SA Congruence?
x
T
A
x
x
y
Not Congruent
S. Stirling
A
T
Not Congruent
A
Not Congruent
Page 1 of 5
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 below, use the given method to try to draw or construct a triangle congruent to the given
triangle, ΔABC ≅ ΔXYZ . Also, try to get two non-congruent triangles! Can you do it? If the triangles have
the same size and shape, no matter what, 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
Copy AC ≅ XZ , AB ≅ XY and CB ≅ ZY . Use a compass.
If the three sides of one triangle are congruent to the three sides of another triangle, then
the triangles are congruent.
B
Y
A
C
Z
X
SAS Congruence Conjecture Does SAS guarantee congruent triangles? YES
Copy AC ≅ XZ , ∠A ≅ ∠X and AB ≅ XY . Use a compass and protractor.
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
Copy ∠ C ≅ ∠ Z , BC ≅ YZ and AB ≅ XY . Use a compass and protractor.
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
Page 2 of 5
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.)
The instructions are the same as on page 2.
ASA Congruence Conjecture Does ASA guarantee congruent triangles? YES
Copy ∠A ≅ ∠X , AC ≅ XZ , and ∠ C ≅ ∠ Z . Use a compass and ruler.
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
Copy ∠A ≅ ∠X and AC ≅ XZ . Use a compass and ruler. Use the m∠ B and m∠ A
to find the m∠ C . Copy ∠ C ≅ ∠ Z .
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.
B
Y
A
Z
X
C
AAA Congruence Conjecture Does AAA guarantee congruent triangles? NO
Copy ∠A ≅ ∠X , ∠B ≅ ∠Y and ∠C ≅ ∠Z . Use a compass and ruler.
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
Complete the Ch 4 Note Sheet, page 9 – 10.
Name __________________________
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 your 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.
ΔTOP
ΔPIT
≅ ΔCong.
TOP
ASA
or AAS
SSA NOT Cong.
Cannot be determined!
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
SAS Cong.
ΔKIE ≅ ΔTIE
ΔABC ≅ ΔXYZ
AAS Cong.
SSS Cong.
ΔSQR ≅ ΔUTR
SSA NOT Cong.
Cannot be determined!
ΔMON ≅ ΔTNO
SAS Cong.
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.
_______
ΔPAT ≅ ΔIMT
ASA OR AAS Cong.
ΔXVW ≅ ΔXZY
ASA OR AAS Cong.
ΔECD ≅ ΔACB
ASA OR AAS Cong.
ΔPQS ≅ ΔPRS
ΔACN ≅ ΔNRA
ΔEQL ≅ ΔGQK
ASA Cong.
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!
Page 5 of 5