Ch 4 Worksheet L1 Rev Key

Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
4.1 Triangles Sum Conjectures
Auxillary line: an extra line or segment that helps you with your proof.
Page 202 Paragraph proof explaining why the Triangle Sum Conjecture is true.
Conjecture: The sum of the measures of the angles in every triangle is 180°.
Given: ∆ABC with auxiliary line
angles labeled as shown.
Show:
EC AB and
m∠ 2 + m∠ 4 + m∠5 = 180°
m∠1 + m∠ 2 + m∠3 = 180° Linear pair conjecture
AC and CB form transversals between parallel lines EC and AB
m∠1 = m∠ 4 and m∠3 = m∠5 because AIA are congruent
Substituting into the first equation above
m∠ 2 + m∠ 4 + m∠5 = 180°
Therefore, the sum of the measures of the angles in every triangle is 180°.
Page 204 #18 Prove Third Angle Conjecture
B
F
Conjecture: If two angles of one triangle are congruent to
two angles of another triangle, then the third angle in each
triangle is congruent to the third angle in the other triangle.
A
Given:
Show:
m∠ A = m∠ E and m∠ B = m∠ F
m∠ C = m ∠ D
C
E
D
m∠ A + m∠ B + m∠ C = 180° and m∠ E + m∠ F + m∠ D = 180° by the
triangle sum conjecture.
Since they both equal 180, m∠ A + m∠ B + m∠ C = m∠ E + m∠ F + m∠ D
Now subtract equal measures m∠ A = m∠ E and m∠ B = m∠ F .
m∠ C = m∠ D Therefore, the third angles are always congruent.
S. Stirling
Page 1 of 20
Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
4.1 Page 203 Exercise #8
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
look 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 2 of 20
Ch 4 Worksheet L1 Rev Key.doc
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.
S. Stirling
Page 3 of 20
B
Ch 4 Worksheet L1 Rev Key.doc
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
∆ABC 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.
S. Stirling
Page 4 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 5 of 20
Ch 4 Worksheet L1 Rev Key.doc
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
∆CAT
C
from
A
A
T
C
T
C
T
Construct
∆FSH
from
F
H
S
H
F
F
S
S
Why were you able to construct ∆CAT but not able to construct ∆FSH ? 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 6 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 7 of 20
Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
EXERCISES Lesson 4.4 Page 224-225 #3 – 10, 12 – 17
Mark diagrams! If congruence cannot be determined, draw a counterexample.
S. Stirling
Page 8 of 20
Ch 4 Worksheet L1 Rev Key.doc
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.
S. Stirling
Page 9 of 20
Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
EXERCISES Lesson 4.5 Page 229-230 #3 – 18
Mark diagrams! If congruence cannot be determined, draw a counterexample.
S. Stirling
Page 10 of 20
Ch 4 Worksheet L1 Rev Key.doc
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.
S. Stirling
Page 11 of 20
Ch 4 Worksheet L1 Rev Key.doc
S. Stirling
Name ___________________________
Page 12 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 13 of 20
Ch 4 Worksheet L1 Rev Key.doc
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
∆AMD ≅ ∆BMC
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
∆ABD ≅ ∆CBD
ASA Congruence
CPCTC or Def. Congruence
S. Stirling
C
Page 14 of 20
Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
4.8 Proving Special Triangle Conjectures
Prove: The bisector of the vertex angle of an isosceles triangle is also the median and altitude to the base.
Given: Isosceles
Prove:
BD
BD
∆ABC
is a median.
is an altitude.
with
AB = BC ; BD bisects ∠ABC
B
DB bisects m∠ABC
given
A
m∠ ABD = m∠ CBD
AB = BC
D
C
BD ≅ BD
def. of angle bisector
given
Shared side
AD = DC
∆ABD ≅ ∆CBD
CPCTC or
Def. Congruence
ASA Cong.
m∠ ADB = m∠ BDC = 90
BD is a median
CPCTC or Def. Congruence
BD is an altitude
Def. of a Median
Def. of an Altitude
Prove: The bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector to the
base.
B
Given: Isosceles
Prove:
BD
∆ABC
with
AB = BC ; BD bisects ∠ABC
is a perpendicular bisector.
BD ≅ BD
DB bisects m∠ABC
given
AB = BC
ASA Congruence
m∠ ADB = m∠ BDC = 90
Linear pair supp. and angles equal.
AD = DC
S. Stirling
C
def. of angle bisector
∆ABD ≅ ∆CBD
CPCTC or
Def. Congruence
D
m∠ ABD = m∠ CBD
Shared side
given
A
BD
is a perpendicular bisector
Def. of a perpendicular bisector
AC ⊥ BD
Def. Perpendicular
Page 15 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 16 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 =
∆TOP ≅ ∆ZAP by AAS Cong.
Cong.
or
or
if you state TO AZ because
alternate interior angles =, then
∠T ≅ ∠Z the lines are parallel.
now ∆TOP ≅ ∆ZAP by ASA Cong.
if you state
∆TRP ≅ ∆APR by
SAS Cong.
∆MSE ≅ ∆OSU by SSS
∠ESM ≅ ∠USO because
vertical angles = , now
∆MSE ≅ ∆OSU by SAS
Cong.
Since ∠GHI ≅ ∠HIG ,
HG ≅ GI because if base
angles =, then isosceles.
∠HGC ≅ ∠IGN because
vertical angles =
∆CGH ≅ ∆NGI by SAS
Cong.
Page 17 of 20
Ch 4 Worksheet L1 Rev Key.doc
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 ___________________________
Since AB CD
∠A ≅ ∠D and ∠B ≅ ∠C
because lines ||, so alternate
interior angles =.
Also ∠BEA ≅ ∠CED because
vertical angles =
∆ABE ≅ ∆DEC 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
∆ACN ≅ ∆OBR or
∆ACN ≅ ∆RBO by SAS
Cong.
∆AMD ≅ ∆UMT 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 18 of 20
Ch 4 Worksheet L1 Rev Key.doc
Since LA TR , ∠A ≅ ∠T
because lines ||, so alternate
interior angles =.
∆SLA ≅ ∆IRT by AAS Cong.
TR ≅ LA Def. of Congruent
Name ___________________________
∆INK ≅ ∆VSE by SSS Cong.
But not needed because
EV = IK and
EV + VI = IK + VI
EI = VK
Triangles or CPCTC
by addition.
Since MN CT ,
Overlapping triangles:
∆ALZ ≅ ∆AIR by ASA
Cong. because
∠MNT ≅ ∠NTC because
lines ||, so alternate interior
angles =.
NT ≅ NT a shared side
Can’t be determined because
you can not get more equal
sides nor angles and SSA does
not guarantee congruence.
S. Stirling
Parts do not match. Both
triangles are AAS but the
angles do not match.
∠A ≅ ∠A same angle.
Since ∠SPT ≅ ∠PTO , the
alternate interior angles = and
lines ||. SP TO
Since ∠OPT ≅ ∠PTS , the
alternate interior angles = and
lines ||. OP TS .
Since the opposite sides are
parallel, STOP is a
parallelogram.
Page 19 of 20
Ch 4 Worksheet L1 Rev Key.doc
Name ___________________________
4.R Page 253 Exercise #27
X
In ∆PCX :
m∠CPX = 30 triangle sum 180 – 30 – 120 = 30
So f larger than a = g
In ∆PXM :
m∠PXM = 60 straight angle 180 – 30 – 90 = 60
m∠PMX = 60 triangle sum 180 – 60 – 60 = 60
So all sides of ∆PXM are equal
f =e=d
In ∆AXM :
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
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