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Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.1 Triangles Sum Conjectures
4.1 Page 203 Exercise #8 Look for overlapping triangles!
Hint: look for large
overlapping triangles (ie.
The one with the 40°, 71°
and a.)
a = 69, b = 47, c = 116,
d = 93, e = 86
4.1 Page 203 Exercise #9
Hint: Fill in angles that do not
have a variable and l ook for
large overlapping triangles!
There are many!!
m = 30, n = 50, p = 82, q = 28,
r = 32, s = 78, t = 118, u = 50
S. Stirling
Page 1 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.2 Group Investigation 1, Base Angles of an Isosceles Triangle
Each of the triangles below is isosceles. Carefully measure the angles of each triangle. (Make sure
the triangles’ angles sum is 180° right?) If you disregard measurement error, are there any patterns
for all isosceles triangles?
A
140
20
B
20
C
A
28
76
B
45
76
45
C
C
90
A
Finish the following conjecture using the vocabulary you learned about isosceles triangles.
Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles are congruent.
Complete the conjecture in the notes and the example problem.
S. Stirling
Page 2 of 18
B
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.2 Group Investigation 2, Is the Converse True?
Write the converse of the Isosceles Triangle Conjecture below.
Converse of the Isosceles Triangle Conjecture
If a triangle has two congruent angles, then it is an isosceles triangle.
Is this converse true? In this investigation, you are going to make congruent angles and then
measure the sides to see if the triangle is isosceles. For each of the following, make
Extend the sides to form
∠A .
Then measure the sides to see if
ΔABC is isosceles.
6.1 cm
6.1 cm
35
35
B
10 cm
C
∠B ≅ ∠C .
C
8.4 cm
70
5.7 cm
8.4 cm
70
B
Is the converse of the Isosceles Triangle Conjecture true? YES
Complete the conjecture in the notes and the example problem.
S. Stirling
Page 3 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.2 Page 209 Exercise #10
Hint: Look for the overlapping triangle involving e, d and 66°.
Do you see 3 equal angles?
a = 124, b = 56, c = 56,
d = 38, e = 38, f = 76,
g = 66, h = 104, k = 76,
n = 86, p = 38
4.2 Page 209 Exercise #11 In the problem, they state that the angles around the center are
congruent. Note: In order for the pattern of tiles to look symmetric, all of the triangles of the same
size must be congruent! How many of the tiles are isosceles triangles?
a = 36, b = 36,
c = 72, d = 108,
e = 36
All of the
triangles are
isosceles.
S. Stirling
Page 4 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.3 Group Investigation 1, Lengths of the Sides of a Triangle
For each of the following, construct the triangle given the three sides. Compare your results with
your group members. When is it possible to construct a triangle from 3 sides and when is it not
possible? Measure the three sides in centimeters. How do the numbers compare?
Construct
ΔCAT
C
from
A
A
T
C
T
C
T
Construct
ΔFSH
from
F
S
H
H
F
F
S
S
Why were you able to construct ΔCAT but not able to construct ΔFSH ? Give more examples
of three side lengths that will NOT make a triangle. Will sides of 4 cm, 6 cm and 10 cm make a
triangle?
Various examples: 2, 5, 9 because 2 + 5 < 9
4, 6 and 10? No because 4 + 6 = 10 NOT a triangle, it’s a segment.
State your observations in the conjecture.
Triangle Inequality Conjecture
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
Complete the conjecture in the notes and the example problems.
S. Stirling
Page 5 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.3 Group Investigation 2, Largest and Smallest Angles in a Triangle
For each of the following triangles, carefully measure the angles. Label the angle with the greatest
measure ∠ L , the angle with the second largest measure ∠ M , and the smallest angle ∠ S .
Now measure the sides in centimeters. . Label the side with the greatest measure l, the side with
the second largest measure m, and the shortest side s.
Which side is opposite ∠ L ? ∠ M ? ∠ S ? Write a conjecture that states where the largest and
smallest angles are in a triangle, in relation to the longest and shortest sides.
M
L
33
l
75
s
S
m
128
19
m
s
L
S
35
70
M
l
Side-Angle Inequality Conjecture
In a triangle, if one side is the longest side, then the angle opposite the longest side is
the largest angle. (And visa versa.)
Likewise, if one side is the shortest side, then the angle opposite the shortest side is
the smallest angle. (And visa versa.)
Does this property apply to other types of polygons? Test it out! Would you really need to measure
these?
∠E is the
E
P
N
A
T
largest angle
but it is
opposite the
shortest side
AT
U
Q
A
D
.
Can’t be true for
polygons with an even
number of sides
because angles are
opposite angles and
sides are opposite
sides.
Complete the conjecture in the notes and the example problems.
S. Stirling
Page 6 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
EXERCISES Lesson 4.4 Page 224-225 #3 – 10, 12 – 17
I
I
Z
P
Z
P
For Exercises 4 – 9, decide whether the triangles are congruent, and name the congruence shortcut you used
(SSS or SAS). If not enough information is given, see if you can use the definitions and conjectures you
have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and
the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot
be determined, write “cannot be determined” and draw a counterexample if possible.
SSS Cong.
FD ≅ FD shared side
SSS Cong.
CV ≅ CV shared side
KA ≅ KA shared side
SSS Cong.
SAS Cong.
Not congruent, sides are not
matched. But, ΔCOT ≅ ΔNAP
by SAS Cong.
AY ≅ YR Def. midpoint
BY ≅ YN If base angles =,
ΔBNY isosceles.
SSS Cong.
Perimeter ΔABC = 180
x + x + 11 + x − 11 = 180
3 x = 180
x = 60
AC = 60 , AB = 60 + 11 = 71
m∠DAE = m∠BAC Vertical angles =
So ΔABC ≅ ΔADE by SAS Cong.
S. Stirling
Page 7 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
For Exercises 12 – 17, if possible, name a triangle congruent to the given triangle and state the congruence
conjecture (SSS or SAS). If not enough information is given, see if you can use the definitions and
conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what
you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If
congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.
ΔANT ≅ ΔFLE
SSS Cong.
Can’t be determined since
SSA is not a congruence
guarantee.
ΔSAT ≅ ΔSAO
Can’t be determined since
the parts do not match up.
One triangle is an ASA the
other is an AAS.
S. Stirling
SAS Cong.
Since AS ≅ AS a shared
side.
ΔGIT ≅ ΔAIN
SSS Cong.
Or if you state ∠GIT ≅ ∠AIN
because vertical angles =
then ΔGIT ≅ ΔAIN by SAS
Cong.
Even with WO ≅ WO a
shared side.
Can’t be determined since
the parts do not match up.
One triangle is an ASA the
other is an AAS.
Page 8 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
EXERCISES Lesson 4.5 Page 229-230 #3 – 18
I
Z
P
I
Z
P
For Exercises 4 – 9, determine whether the triangles are congruent, and name the congruence shortcut you
used (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and
conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what
you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If
congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.
∠AMD ≅ ∠RMC because
vertical angles =
then ΔAMD ≅ ΔRMC by
ASA Cong.
∠HOW ≅ ∠FEW because
vertical angles =
Even with this there is not
enough information to
determine congruence.
S. Stirling
Even with FS ≅ FS a
shared side.
Can’t be determined.
This would be a SSA
which does not guarantee
congruence.
ΔBOX ≅ ΔCAR
ASA Cong.
ΔGAS ≅ ΔIOL
AAS Cong.
Or if you use the third angle
conjecture: state ∠S ≅ ∠L ,
then ΔGAS ≅ ΔIOL by ASA
Cong.
∠INT ≅ ∠TLA and
∠NIT ≅ ∠TAL because lines ||,
so alternate interior angles =.
∠NTI ≅ ∠LTA because vertical
angles =
Even with all of this the triangles
are not necessarily congruent
because AAA does not guarantee
congruence.
Page 9 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence
conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions
and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write
what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If
congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.
ΔFAD ≅ ΔFED
SSS Cong.
Since FD ≅ FD a shared
side.
Overlapping Triangles
ΔPOE ≅ ΔPRN by ASA
Cong.
and
∠OSN ≅ ∠RSE because
vertical angles =
ΔSON ≅ ΔSRE by ASA
Cong.
S. Stirling
∠H ≅ ∠T and
∠O ≅ ∠A because lines ||, so
alternate interior angles =.
ΔWHO ≅ ΔWTA by AAS Cong.
or…
∠HWO ≅ ∠YWA because
vertical angles =
∠H ≅ ∠T because lines ||, so
alternate interior angles =.
ΔWHO ≅ ΔWTA by ASA Cong.
∠DMA ≅ ∠RMC because
vertical angles =
Still can’t be determined
since the parts do not match
up. One triangle is an ASA
the other is an AAS.
∠LAT ≅ ∠SAT def. of
angle bisector
Since TA ≅ TA a shared side,
ΔLAT ≅ ΔSAT by SAS
Cong.
Overlapping Triangles:
MR ≅ MR a shared side.
ΔRMF ≅ ΔMRA by SAS
Cong.
Page 10 of 18
Ch 4 Worksheet L2 Key 3 post
∠B ≅ ∠K and ∠L ≅ ∠C
because lines ||, so alternate
interior angles =.
∠BAL ≅ ∠KAC because
vertical angles =
Even with all of this the triangles
are not necessarily congruent
because AAA does not guarantee
congruence.
Name ___________________________
WL ≅ WL a shared side.
ΔLAW ≅ ΔWKL by ASA
Cong.
Perimeter ΔABC = 138
x + x + 4 + x − 4 = 138
3 x = 138
x = 46
AC = 46
m∠DAE = m∠BAC Vertical angles =
∠E ≅ ∠C and ∠D ≅ ∠B because
lines ||, so alternate interior angles =.
So ΔABC ≅ ΔADE by AAS Cong. or by
ASA Cong.
S. Stirling
Page 11 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.4 Page 226 Exercise #23
a = 37, b = 143, c = 37, d = 58
e = 37, f = 53, g = 48, h = 84,
k = 96, m = 26, p = 69, r = 111,
s = 69
4.6 Page 234 Exercise #18
a = 112, b = 68, c = 44,
d = 44, e = 136, f = 68,
g = 68, h = 56, k = 68,
l = 56, m = 124
S. Stirling
Page 12 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.6 Corresponding Parts of Congruent Triangles
4.6 Page 232 Example A
Given:
Prove:
C
AM ≅ MB and m∠A = m∠B
AD ≅ BC
B
2
1
M
A
D
m∠ A = m∠ B
AM ≅ MB
given
given
m∠1 = m∠ 2
ΔAMD ≅ ΔBMC
Vertical angles =
ASA Congruence
AD ≅ BC
CPCTC or Def. Congruence
B
Example B
Given:
Prove:
BD ⊥ AC and DB bisects m∠ABC
∠A ≅ ∠C
m∠ADB = m∠BDC = 90
BD ⊥ AC
given
A
D
def. of perpendicular
DB bisects m∠ABC
given
m∠ABD = m∠CBD
BD ≅ BD
Shared side
def. of angle bisector
∠A ≅ ∠C
ΔABD ≅ ΔCBD
ASA Congruence
CPCTC or Def. Congruence
S. Stirling
C
Page 13 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.7 Page 241 Exercise #13
a = 72, b = 36, c = 144, d = 36
e = 144, f = 18, g = 162, h = 144,
j = 36, k = 54, m = 126
4.8 Page 247 Exercise #12
a = 128, b = 128, c = 52, d = 76
e = 104, f = 104, g = 76, h = 52,
j = 70, k = 70, l = 40, m = 110,
n = 58
S. Stirling
Page 14 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
EXERCISES Ch 4 Review Page 252 #7 – 24
For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence
conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions
and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write
what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If
congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible.
The triangles are not
necessarily congruent
because SSA does not
guarantee congruence.
Can’t be determined
because you can not
get more equal sides
nor angles and SSA
does not guarantee
congruence.
S. Stirling
∠OPT ≅ ∠APZ vertical angles =
ΔTOP ≅ ΔZAP by AAS Cong.
Cong.
or
or
if you state TO & AZ because
alternate interior angles =, then
∠T ≅ ∠Z the lines are parallel.
now ΔTOP ≅ ΔZAP by ASA Cong.
if you state
ΔTRP ≅ ΔAPR by
SAS Cong.
ΔMSE ≅ ΔOSU by SSS
∠ESM ≅ ∠USO because
vertical angles = , now
ΔMSE ≅ ΔOSU by SAS
Cong.
Since ∠GHI ≅ ∠HIG ,
HG ≅ GI because if base
angles =, then isosceles.
∠HGC ≅ ∠IGN because
vertical angles =
ΔCGH ≅ ΔNGI by SAS
Cong.
Page 15 of 18
Ch 4 Worksheet L2 Key 3 post
If isosceles, then base angles
=. So ∠O ≅ ∠T .
WH ≅ WH a shared side
Can’t be determined because
you can not get more equal
sides nor angles and SSA does
not guarantee congruence.
Name ___________________________
HJJG HJJG
Since AB & CD
∠A ≅ ∠D and ∠B ≅ ∠C
because lines ||, so alternate
interior angles =.
Also ∠BEA ≅ ∠CED because
vertical angles =
ΔABE ≅ ΔDEC by AAS Cong.
or ASA Cong.
Since it is a regular
polygon all sides and
angles are =:
∠C ≅ ∠B and
CN = CA = OB = BR
So
ΔACN ≅ ΔOBR or
ΔACN ≅ ΔRBO by SAS
Cong.
ΔAMD ≅ ΔUMT by SAS
Cong.
AD ≅ UT Def. of
Congruent Triangles or
CPCTC
S. Stirling
Can’t be determined
because you can not
get more equal sides
nor angles and AAA
does not guarantee
congruence.
Can’t be determined
because you can not
get more equal sides
nor angles and SSA
does not guarantee
congruence.
Page 16 of 18
Ch 4 Worksheet L2 Key 3 post
Since LA & TR , ∠A ≅ ∠T
because lines ||, so alternate
interior angles =.
ΔSLA ≅ ΔIRT by AAS Cong.
TR ≅ LA Def. of Congruent
Triangles or CPCTC
Since MN & CT ,
∠MNT ≅ ∠NTC because
Name ___________________________
ΔINK ≅ ΔVSE by SSS Cong.
But not needed because
EV = IK and
Parts do not match. Both
triangles are AAS but the
angles do not match.
EV + VI = IK + VI
EI = VK
by addition.
Overlapping triangles:
ΔALZ ≅ ΔAIR by ASA
Cong. because
Since ∠SPT ≅ ∠PTO , the
alternate interior angles = and
NT ≅ NT a shared side
lines ||. SP & TO
Since ∠OPT ≅ ∠PTS , the
alternate interior angles = and
Can’t be determined because
you can not get more equal
sides nor angles and SSA does
not guarantee congruence.
lines ||. OP & TS .
Since the opposite sides are
parallel, STOP is a
parallelogram.
lines ||, so alternate interior
angles =.
S. Stirling
∠A ≅ ∠A same angle.
Page 17 of 18
Ch 4 Worksheet L2 Key 3 post
Name ___________________________
4.R Page 253 Exercise #27
X
In ΔPCX :
m∠CPX = 30 triangle sum 180 – 30 – 120 = 30
So f larger than a = g
In ΔPXM :
m∠PXM = 60 straight angle 180 – 30 – 90 = 60
m∠PMX = 60 triangle sum 180 – 60 – 60 = 60
So all sides of ΔPXM are equal
f =e=d
In ΔAXM :
m∠XMA = 45 triangle sum 180 – 90 – 45 = 45
Since base angles =, triangle is isosceles. So
So c larger than d = b
So c is the largest overall!
S. Stirling
Page 18 of 18