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Foundations of Math 2
Unit 3 : Triangles
Notes & Homework
Mrs. Bello
Fall 2016
1
Foundations Math II
Name_____________________ Pd ____
Unit 3: Triangles
Day
1
Lesson
Homework
Stamp
Triangle Sum
2
Isosceles Triangle
3
Isosceles Triangle
4
Dilations & Similarity
5
Quiz #1
6
Mid-segments
7
Triangle Congruence
8
Triangle Congruence
9
Quiz #2
10
Flow Charts Proofs
11
Flow Charts Proofs
12
Review
13
Test #3 – Triangles
2
Foundations of Math 2 – Unit 3: Triangles
Day 1: Triangle Sum Theorem
Warm-Up
Grab a colored sheet of paper. I want you to use a ruler and draw a triangle.
Label the interior (inside) angles (∠1, ∠2, ∠3). Now cut your triangle out.
Now we are going to tear the 3 corners of our triangle off. I want you to line them up on the following line.
What do you notice?
Triangle Sum Theorem: _________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
3
You try!
Solve for the missing angle.
1.
2.
Solve for x.
3.
4.
Linear Pairs: ________________________________________________________________________________
___________________________________________________________________________________________
x
76o
You try!
Find x, then y.
y
x
126
o
Vertical Angles: _____________________________________________________________________________
___________________________________________________________________________________________
4
Practice:
Name____________________
5
6
Day 2 – Isosceles Triangles
Parts of a Triangle:
Triangle – a three-sided polygon
Name –
Sides –
Vertices –
Angles –
Classifying Triangles by Angles:
Acute ∆
Obtuse ∆
Right ∆
Isosceles ∆
Equilateral ∆
Equiangular ∆ -
Classifying Triangles by Sides:
Scalene ∆
Example #1: Find x and the measure of each side of equilateral triangle RST.
7
Example #2: Find x, JM, MN, and JN if ∆JMN is an isosceles triangle
with JM  MN .
Isosceles Triangle: A triangle with at least __________ sides congruent.
Isosceles Triangle Theorem: If two sides of a triangle are ____________________, then the angles opposite those sides
are ____________________.
Ex:
Example #3: If DE  CD , BC  AC , and mCDE  120 , what is the measure of BAC ?
8
Theorem:
If two angles of a _______________ are congruent, then the sides opposite those angles are
___________________.
Ex:
Example #4:
a.) Name all of the congruent angles.
b.) Name all of the congruent segments.
A triangle is _____________________ if and only if it is ___________________.
Each angle of an equilateral triangle measures ________.
Example #5: ∆EFG is equilateral, and EH bisects E .
a.) Find m1 and m2 .
b.) Find x.
9
10
Day 3: Isosceles Triangles cont.
Warm-Up
1.
2.
32o
70o
a
60o
x
b
100o
3. (x,y)(x+2, -y)
c
d
e f
4. Reflect over y= -x.
A
A
C
C
B
A’_________ B’_________ C’__________
B
A’_________ B’_________ C’__________
11
Find x and y
1)
2)
3)
4
3
4
4
3
4)
5) Equilateral Triangle
4x
40
60
6) Equilateral Triangle
10
y
Find x, y and z
7)
8)
5z
9)
9
9
3x
4y
9
12
Find x (and y) in each figure below.
1.
2.
3.
x
x
4.
130o
5.
6.
13
14
Day 4: Dilation and Similarity
Warm-Up
3. Dilate the figure with respect to the origin by a scale factor
of ½. List the preimage and image points.
4. Dilate the figure with respect to the origin by a scale factor
of 1/4 . List the preimage and image points.
15
Similarity Practice
Solve for x.
1. If
10.
x 8
x y
 , then
 .
y 3
8 3
x
9

6 24
2. If
11.
x 8
x 8 y 3
 , then

.
y 3
8
3
4
3

y 3 y 3
12.
5
3

2y  7 y  3
Solve for the variable.
13. MN:MO is 3:4
14. PQR side lengths: STU side lengths is 1:3
M
x
P
S
5
x
9
R
O
N
U
36
12
Q
T
Use the diagram and the given information to find the unknown length.
15. Given
AB AE

, find BC.
BC ED
16. Given
AB AE

, find BC.
BC ED
.
16
Notes: Similar Polygons
Similar Polygons:
SYMBOL for SIMILAR: ________
Corresponding angles are _____________________________________________
Corresponding sides are _______________________________________________
Writing Similarity Statements:
Corresponding <’s:
Proportional Sides:
A BC
XYZ



If 2 polygons are _____________, then the ratio of the lengths of 2 corresponding
sides is called the ___________________.
What is the scale factor of ∆ABC to ∆XYZ? ________________
Practice:
1.) If polygon LMNO ~HIJK , completing proportions and congruence statements.
Hint: Draw a
diagram!!
a. M  __?__
d.
MN

IJ
?
JK
b. K  __?__
e.
HK
HI

?
LM
c. N  __?__
f.
IJ
HK

MN
?
17
2.) In the diagram, polygon ABCD ~ GHIJ.
A
11
D
8
x
x
B
G
y
H
5.5
11
C
a. Find the scale factor of polygon
ABCD to polygon GHIJ.
c. Find the values of x and y.
J
8
I
b. Find the scale factor of polygon
GHIJ to polygon ABCD.
d. Find the perimeter of each polygon.
e. Find the ratio to the perimeter of ABCD to perimeter of GHIJ.
If 2 polygons are ___________, then the ratio of their perimeters
is equal to the ratios of their ___________.
3.)
The ratio of one side of ∆ABC to the corresponding side of similar ∆DEF
is 3:5. The perimeter of ∆DEF is 48in. What is the perimeter of ∆ABC?
18
Classwork (# 1 – 9)
Show proportions for all problems.
1. Parallelogram EFGH is similar to parallelogram WXYZ.
X
F
G
Y
6 in
2 in
E
3 in
H
W
Z
What is the length of WZ ?
a) 3 in
b) 6 in
c) 7 in
d) 9 in
2. Lance the alien is 5 feet tall. His shadow is 8 feet long.
5 ft
32 ft
8 ft
At the same time of day, a tree’s shadow is 32 feet long. What is the height of the tree?
a) 20 feet
b) 24 feet
c) 29 feet
d) 51 feet
X
?
3. Pentagon JKLMN is similar to pentagon VWXYZ.
What is the measurement of angle X?
L
a) 30
b) 60
c) 150
60
K 150
4. Triangle LMN is similar to triangle XYZ.
Y
W
d) 120
J
M
N V
Z
19
L
Z
18 feet
24 feet
8 feet
Y
N
12 feet
X
M
What is the length of YX ?
a) 2 feet
b) 3 feet
c) 4 feet
d) 6 feet
5. Triangle PQR is similar to triangle DEF as shown.
E
Q
6 cm
4 cm
P
6 cm
R
D
9 cm
F
Which describes the relationship between the corresponding sides of the two
triangles?
a)
PQ 4

DE 6
b)
PQ 6

DE 4
c)
PQ 4

EF 9
d)
PR 6

DE 6
1
6. A six-foot-tall person is standing next to a flagpole. The person is casting a shadow 1 feet
2
in length, while the flagpole is casting a shadow 5 feet in length. How tall is the flagpole?
a) 30 ft
b) 25 ft
c) 20 ft
d) 15 ft
20
7. PQR is similar to XYZ.
Y
Q
6
5
30
R
P
Z
10
X
What is the perimeter of XYZ?
a) 21 cm
b) 63 cm
c) 105 cm
d) 126 cm
8. The shadow cast by a one-foot ruler is 8 inches long. At the same time, the shadow cast by a
pine tree is 24 feet long.
a) 3 feet
b) 16 feet
c) 36 feet
d) 192 feet
1 foot
What is the height, in feet, of the pine tree?
8 inches
9. If triangles ADE and ABC shown in the figure
to the right are similar, what is the value of x?
24 feet
12
C
B
4
a) 4
b) 5
c) 6
d) 8
e) 10
x
E
D
2
3
A
21
Day 5 (Quiz Day) : Mid-segments
Warm-Up
.
e
45
44
a=______
b=______
c=______
d=______
e=______
f=______
g=______
h=______
d
a
51 b
f
c
g
45
h
22
23
24
25
26
27
Day 6: Midsegments cont.
Warm-Up
a=_________
1.
2.
31
x=_______
b=_________
a
109
b c=_________
d=_________
2x-9
36
46
3x+8
c e=_________
d
19
e
3.
e
39 
45
o
36
44
o
a=______
b=______
c=______
d=______
e=______
f=______
g=______
h=______
d
48
a
o
51 b
f
c
45
51o
g
h
28
1.
x
18
2.
5x
70
3.
84
3x
4.
x-1
45
5.
5x-2
4
29
For 6-11, name the segment that is parallel to the given segment.
A
F
C
6. AB
7. EF
8. GE
9. BC
10. CA
11. FG
G
E
B
12. Find the values of x and y.
3x-6
y
x
2x+1
30
Day 7: Triangle Congruence
31
Congruent Figures: have the same __________________ & same _______________________.
Each _____________________ (“matching”) side and angle of congruent figures will also be _______.
Example #1:
(Points can be named in any consecutive order, but each
corresponding vertex must be in the same order for each figure).
ABCDE ≅ VWXYZ
Congruent Angles
Congruent Sides
Example #2: Given: ABCD≅ EFGH. Complete the following
a) Rewrite the congruence statement in at least 2 more ways.
b) Name all congruent angles
c) Name all congruent sides
We will deal mostly with congruent triangles. Two triangles are congruent if and only if their vertices can be matched up
so that the _______________________________ (both angles & sides) are congruent.
Example #3: Given: ∆𝐻𝐼𝐽 ≅ ∆𝑀𝑁𝑂. Name all congruent sides and all congruent angles.
Write the congruence statement in at least 2 more ways.
32
Example #4: If ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍, and 𝑚∠𝐴 = 3𝑥 + 12, 𝑚∠𝐵 = 3𝑥 + 1 and 𝑚∠𝑋 = 𝑥 + 44, find x.
What if we wanted to prove that 2 polygons were congruent? What would we need to do?
Because triangles only have three sides, we can take some shortcuts…
I. If all three sides are given, we call this ___________.
SSS Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are
congruent.
II. If 2 sides and the angle BETWEEN those sides are given, we call this _________.
SAS Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle
of another triangle, then the triangles are congruent.
Included means _______________
III. If 2 angles and the side BETWEEN those angles are given, we call this _________.
ASA Postulate: If 2 angles and the included side are congruent to 2 angles and the include side of another
triangle, then the triangles are congruent.
IV. If 2 angles and the side NOT BETWEEN those angles are given, we call this _______.
AAS Postulate: If 2 angles and their non-included side are congruent to 2 angles and the non-included side of
another triangle, then the triangles are congruent.
33
Example #5: Are the triangles congruent? If so, why?
Anytime that 2 triangles share a side, think _________________ property!
̅̅̅̅, EF
̅.
̅ ≅ GH
̅̅̅̅ ≅ HF
̅̅̅̅ and G is the midpoint of GI
Example #6: IE
34
35
Triangle Congruence Picture Questions
I.
If the triangles can be proven congruent, give the reason (SSS, SAS, ASA, or AAS). If there is not enough
information to prove the triangles congruent, write “none.”
1.
5.
9.
6.
10.
7.
11.
8.
12.
2.
3.
4.
II.
Determine whether you can conclude that another triangle is congruent to ∆ABC.
 If so, complete the congruence statement and give the reason (SSS, SAS, ASA, or AAS).
 If not, write “none.”
A
1.
A
C
C
2.
B
B
K
D
3.
P
B
N
Y
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
C
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
A
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
36
4.
A
B
5.
S
X
B
6.
A
C
A
Z B
Y
C
J
C
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
7.
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
8.
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
9.
P
A
C
P
A
60
B
61
C
B
C
B
Q
D
60
A
Q
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
∆𝐴𝐵𝐶 ≅ ∆_____________
by ______________
For Question #1-4, 𝜟𝑭𝑰𝑵 ≅ 𝜟𝑾𝑬𝑩
1. Name the three pairs of corresponding sides. _______________
2. Name the three pairs of corresponding angles ______________
3. Is it correct to say 𝑭 ≅ 𝜟𝑩𝑬𝑾 ?_____ Explain why?_________________________________________
4. Is it correct to say 𝑭 ≅ 𝜟𝑬𝑩𝑾 ?_____ Explain why?_________________________________________
For questions #5-9, use the congruent triangles to the right
5. 𝛥𝐴𝐵𝑂 ≅ 𝛥________
6. ∡𝐴 ≅ ______
̅̅̅̅ ≅ ______
7. 𝐴𝑂
8. 𝐵𝑂 = ______
C
D
O
A
B
37
What additional information is required in order to
know that the triangles are congruent by the given
reason?
D
1. ASA
5. SAS
R
I
J
U
T
S
2.
SAS
X
6. ASA
K
S
L
W
S
T
H
M
V
T
K
U
3. SAS
C
7. SSS
L
R
Q
B
K
A
S
J
T
4. ASA
8. SAS
D
F
U
J
L
W
K
V
M
E
K
38
Day 8: Triangle Congruence cont.
Warm-Up
Draw and label a diagram. Then solve for the variable and the missing measure or length.
1. If ∆𝐵𝐴𝑇 ≅ ∆𝐷𝑂𝐺, and 𝑚∠𝐵 = 14, 𝑚∠𝐺 = 29, 𝑎𝑛𝑑 𝑚∠𝑂 = 10𝑥 + 7. Find the value of x 𝑚∠𝑂.
x = ___________
𝑚∠𝑂= _________
2. If ∆𝐶𝑂𝑊 ≅ ∆𝑃𝐼𝐺, and 𝐶𝑂 = 25, 𝐶𝑊 = 18, 𝐼𝐺 = 23, 𝑎𝑛𝑑 𝑃𝐺 = 7𝑥 − 17 . Find the value of x and PG.
x = ___________
PG=___________
3. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅 and 𝐷𝐸 = 3𝑥 − 10, 𝑄𝑅 = 4𝑥 − 23, 𝑎𝑛𝑑 𝑃𝑄 = 2𝑥 + 7. Find the value of x and EF.
x = ___________
EF = __________
39
I.
Use the given information and triangle congruence statement to complete the following.
1. ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐸𝑂, AB = 4, BC = 6, and AC = 8.
What is the length of ̅̅̅̅
𝐺𝑂? How do you know?
2. ∆𝐵𝐴𝐷 ≅ ∆𝐿𝑈𝐾, 𝑚∠𝐷 = 52°, 𝑚∠𝐵 = 48°, 𝑎𝑛𝑑 𝑚∠𝐴 = 80°.
a. What is the largest angle of ∆𝐿𝑈𝐾?
b. What is the smallest angle of ∆𝐿𝑈𝐾?
3. ∆𝑆𝑈𝑁 ≅ ∆𝐻𝑂𝑇.
or why not.
∆𝑆𝑈𝑁 is isosceles. Is there enough information to determine if ∆𝐻𝑂𝑇 is isosceles? Explain why
40
II.
1.
Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able
to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in
the blanks.
∆𝐸𝐹𝐷 ≅ ∆___________
∆𝐴𝐵𝐶 ≅ ∆___________
2.
by ________
3. ∆𝐿𝐾𝑀 ≅ ∆___________
by ________
E
by ________
L
A
D
F
B
H
C
K
M
T
G
J
4. ∆𝐴𝐵𝐶 ≅ ∆___________
∆𝐴𝐵𝐶 ≅ ∆___________
5.
by ________
A
by ________
A
X
D
C
C
E
Y
B
Use the given information to mark the diagram
and any additional congruence you can determine from the
diagram. Then complete the triangle congruence statement and give the reason for triangle congruence .
B
III.
1.
C
1
2.
A
2
4
D
B
3
B
A
Given: ∠1 ≅ ∠3,
∠2 ≅ ∠4
Given: ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷,
∆𝐴𝐵𝐶 ≅ ∆__________ by __________
3.
C
D
F
∆𝐴𝐵𝐷 ≅ ∆__________ by __________
A
4.
C
1
A
2
G
E
B
∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵
D
4
3
B
̅̅̅̅
̅̅̅̅ 𝑎𝑛𝑑 𝐸𝐴
Given: 𝐺 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐹𝐵
Given: ∠1 ≅ ∠3,
̅̅̅̅ ≅ 𝐴𝐵
̅̅̅̅
𝐶𝐷
∆𝐴𝐵𝐺 ≅ ∆__________ by __________
∆𝐴𝐵𝐶 ≅ ∆__________ by __________
41
Congruent Triangles Activity
EXAMPLE:
1. Name the congruent triangles Δ_ABC_≅Δ_DEF_
2. The triangles are congruent by the _ASA_ postulate
∡𝑪 ≅ ∡𝑭
5. Draw the congruent triangles
A
3. List all of the congruent, corresponding sides
̅̅̅̅ ≅ ̅̅̅̅
𝑨𝑩
𝑫𝑬
̅̅̅̅ ≅ 𝑬𝑭
̅̅̅̅
𝑩𝑪
̅̅̅̅ ≅ 𝑫𝑭
̅̅̅̅
𝑨𝑪
4. List all of the congruent, corresponding angles
∡𝑨 ≅ ∡𝑫
∡𝑩 ≅ ∡𝑬
C
D
F
B
E
1. Name the congruent triangles Δ_OSU_≅Δ_BCK_
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
5. Draw the congruent triangles
4. List all of the congruent, corresponding angles
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _SAS_ postulate
3. List all of the congruent, corresponding sides
5. Draw the congruent triangles
4. List all of the congruent, corresponding angles
42
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
̅̅̅̅̅ ≅ 𝒁𝑰
̅̅̅
𝑴𝑹
̅̅̅̅̅
̅̅̅̅
𝑳𝑴 ≅ 𝑴𝑰
̅̅̅̅
𝑳𝑹 ≅ ̅̅̅̅̅
𝑴𝒁
4. List all of the congruent, corresponding angles
5. Draw the congruent triangles
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
∡𝑬 ≅ ∡𝑫
5. Draw the congruent triangles
4. List all of the congruent, corresponding angles
∡𝑸 ≅ ∡𝑨
∡𝑾 ≅ ∡𝑺
1. Name the congruent triangles Δ_____≅Δ_____
2. The triangles are congruent by the _____ postulate
3. List all of the congruent, corresponding sides
4. List all of the congruent, corresponding angles
5. Draw the congruent triangles
F
I
A
X
L
43
Day 9 (Quiz Day) : Triangle Congruence Cont.
For questions #1-3, use the congruent pentagons below
K
S
E
C
4 cm
B
5. B corresponds to _____
6. BLACK ≅ __________
5 cm
R
7. KB = ______ cm
A
L
H
̅̅̅̅ ⊥ 𝐿𝐴
̅̅̅̅, name two right angles
8. If 𝐶𝐴
4.5 cm
O
from the figures __________
9. If ΔDEF ≅ ΔRST, 𝑚∡𝐷 = 100 and 𝑚∡𝐹 = 40, name four congruent angles. _______________________
I.
Δ𝑃𝑄𝑅 ≅ Δ𝐴𝐵𝐶. Find the values of x and y.
1. 𝑚∠𝑅 = 5𝑥 + 70, 𝑚∠𝐶 = 24𝑥 − 25, 𝑄𝑅 = 4𝑦 + 2, 𝐵𝐶 = 6𝑦 − 8
2. 𝑚∠𝑅 = 90 − 𝑦, 𝑚∠𝐶 = 13, 𝑃𝑅 = 5𝑥 − 10 , 𝐴𝐶 = 32 − 𝑥
3. 𝑃𝑄 = 5𝑥 − 31, 𝐴𝐵 = −3𝑥 + 1 , 𝑄𝑅 = −3𝑦 − 1, 𝐵𝐶 = −2𝑦 − 2
4. 𝑚∠𝐴 = 15𝑦 − 3, 𝑚∠𝑃 = 12𝑦 + 30, 𝑃𝑄 = 11 − 𝑥, 𝐴𝐵 = 11𝑥 − 1
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5. 𝐴𝐵 = 2𝑥, 𝑃𝑄 = 18, 𝐵𝐶 = 11, 𝑄𝑅 = 4𝑦 + 3
6. Δ𝑋𝑌𝑍 ≅ Δ𝑀𝑁𝑂, 𝑚∠𝑋 = 𝑥 + 10, 𝑚∠𝑀 = 4𝑥 − 17, 𝑚∠𝑌 = 2𝑦, and 𝑚∠𝑁 = 𝑦 + 6. Find 𝑚∠𝑋 and 𝑚∠𝑌.
II. Solve.
1. The perimeter of ABCD is 85. Find the value of x. Is Δ𝐴𝐵𝐶 congruent to Δ𝐴𝐷𝐶? Explain.
C
4x
6x - 11
B
D
5x - 7
3x + 4
A
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Day 10: Flow Chart Proofs
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E is the
midpoint
of ̅̅̅̅
𝐵𝐷

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Use a Flow Chart to prove each of the following triangles congruent.
1. Given
BAD 
Prove  ABD  
BCD
CBD
2. Given:
B 
E
3. Given: AC  DC
C is the midpoint of BE & DA Prove:  CAB   CDB
Prove:  ABC   DEC
4. Given: AD  BC
BA  CD
Prove:  ABC   CDA
5. Why is this situation not
possible to prove?
6. Given BC  EF
CA  FD
Prove:
 BCA   EDF
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1)
2)
3)
4)
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4)
6)
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Day 11: Flow Proof Charts Cont.
Class work
Prove:  ADB   CDB
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Day 12 – Unit 3 Study Guide: Triangles
For # 1-3, describe each triangle by sides and angles. Use the word bank provided.
Acute ∆ , right ∆ , Obtuse ∆ , scalene ∆ , isosceles ∆ , equilateral ∆
1.
2.
a. ________________
a. ________________
b. ________________
b. ________________
3.
a.______________________
b. ______________________
For # 4-6, find the given variable(s) for each of the pictures below.
4. t= ___
5.
a = ___
55
12t+8
72
8t-3
4a+12
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6. y = ___
30
7y – 10
y
7. Solve for x, y, and z.
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x= _______
y = ________
z = _______
2z - 27𝑜
x-6
y + 13
z + 15𝑜
x + 10
For # 8 – 13, state whether the triangles are congruent by : SSS, SAS, ASA, AAS, or not possible.
8.
9.
10.
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Name the congruent angles and sides for each pair of congruent triangles.
14. Given Δ ABC ≅ ΔDEF
____________
_____________
____________
____________
_____________
_____________
For #15-19, find the value of x.
15.
16.
2x - 11
15
M
3x+12
x = ______
17.
x + 48
x = ______
18.
x
4x-12
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2x + 30
x = ______
x = ______
19.
80
45
50
70
x = ______
x
***All tests are cumulative! Don’t forget to
review Unit 1 material on Transformations!
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