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