Download Name: Period: ______ 4.1 Tricky Triangles 1) Triangle Sum

Name: ____________________________ Period: ________
4.1 Tricky Triangles
1) Triangle Sum Conjecture: The sum of the measures of the angles in every triangle is ___________.
2) Determine the missing angle in each diagram
a
a = ______
b
b= ______
570
360
c= ______
c
d
380
120
e
g 1240
f
380
h
i
1380
750
d= ______
450
e= ______
h= ______
f= ______
i= ______
g= ______
j = ______
3)
Snyder Rd
k
m
Prince Rd
n
City Designers planned many Tucson streets at right angles. Houghton Rd is perpendicular to both
Snyder Rd and Prince Rd. Melpomene is perpendicular to Prince and Snyder Roads.
a) What can you prove about Snyder Rd and Prince road according to the given information? Explain.
b) The angle Prince Rd makes with Catalina Highway is a 440 angle. Find the unknown angles
k= ______ m= ______ n= ______
line m
4)
2
3
4
1
line n
a) Given: m || n
Prove: The sum of the angles of ΔABC is 1800
5
Name: _______________________________ Period______4.2 Classifying Triangles/Exterior Angle Theorem
1. Use the diagram indicated to prove the exterior angle theorem. Your givens come from the diagram.
The conclusion of your proof should say that 𝑚∠1 = 𝑚∠2 + 𝑚∠3
Statements
Reasons
Questions 2-5: Classify each triangle by the angles, and sides. Assume that the only given information are
the congruence marks, and angle indicators.
Sketch an example of the type of triangle described. Mark the triangle to indicate what information is
known. If no triangle can be drawn, write “not possible.”
6) acute isosceles
7) right scalene
8) right isosceles
9) right equilateral
10) acute scalene
11) obtuse scalene
12) right obtuse
13) equilateral
12) acute equilateral
Questions 13- 18 Find the measure of each indicated angle (?).
13)
14)
15)
16)
17)
18)
Name: _________________________
Period: ________
For 1 & 2, shade a different triangle in each image.
1)
2)
4.3 Overlapping Triangles
For 3 & 4, copy the diagram as many times as needed to shade all the different triangles in each image.
3)
4)
5) Draw your own shape that is made up of at least 4 overlapping triangles. Then recopy your design as
many times as needed to shade all the different triangles in your image.
Name: ___________________________
Period: _________
4.4 Triangle Inequality Problem Set
𝑥+𝑦 >𝑧
Three numbers are given as the side lengths of a triangle. Use the triangle inequality to determine whether
such a triangle can exist
1) 7, 5, 4
2) 3, 6, 2
3) 5, 2, 4
4) 8, 2, 8
5) 9, 6, 5
6) 5, 8, 4
7) 4, 7, 8
8) 11, 12, 9
9) 3, 10, 8
10) 1, 13, 13
11) 2, 15, 16
12) 10, 18, 10
Two side lengths of a triangle are given, determine the range of values that are possible for the 3rd side.
13) 9, 5
14) 5, 8
15) 6, 10
16) 6, 9
17) 11, 8
18) 14, 11
Use your compass and a straight edge to draw a triangle given each set of measurements (label):
19) 7cm, 5cm, 4 cm
20) 8 cm, 8cm, 2cm
21) 3 cm, 6 cm, 2 cm
22) 6 cm, 6 cm, 6 cm
Name: _________________________
Period: ________
4.6 SSS and SAS
SSS: Three sides of one triangle are congruent to the corresponding sides of another triangle
SAS: Two sides and the included angle of one triangle are congruent to the corresponding
parts of another triangle
Identify which property will prove these triangles congruent, SSS or SAS. If neither method works say
“neither”
1.
2.
3.
4.
5.
6.
7.
8.
Sate what additional information is required in order to know that the triangles are congruent
FOR THE GIVEN REASON. Remember ORDER matters. Write the triangle congruency.
Example:
SAS
Since one side is marked 𝐻𝐼 ≅ 𝐾𝐼 and from the diagram there
is a pair of vertical angles so ∠𝐻𝐼𝐽 ≅ ∠𝐾𝐼𝑀, so we need
One more piece of needed information: ___________________
To prove ∆𝐻𝐼𝐽 ≅ ∆𝐾𝐼𝑀 𝑏𝑦 𝑆𝐴𝑆
9.
prove by: SSS
One more piece of needed information: ___________________
Δ KMH ≅ Δ ________
10.
C
A
prove by: SAS
T
F
One more piece of needed information: ___________________
Δ CAT ≅ Δ ________
11.
prove by: SAS
One more piece of needed information: ___________________
Δ MKL ≅ Δ ________
12.
prove by: SSS
One more piece of needed information: ___________________
Δ XZY ≅ Δ ________
4.7 Geometry
Name___________________________________
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ASA and AAS Congruence
Period____
State if the two triangles are congruent. If they are, state how you know.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Worksheet by Kuta Software LLC
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11. Use the figure to determine which (if any) triangles
are congruent to one another.
12. Determine if
. Explain your reasoning.
13. In the figure,
is an equilateral triangle, does that mean
is also equilateral? Explain your reasoning.
Name: ______________________________Period: _______
4.8 Congruence Shortcuts that Fail
1. Demonstrate (By construction) that AAA doesn’t produce two congruent triangles.
2. Using an angle of 45 degrees, and side lengths
of 4cm, and 3.5cm, show that SSA will produce
two different triangles which are not congruent.
The diagram on the right shows the same setup
except with an angle of 30 degrees and side
lengths 10 and 6.
Name two short cuts that don’t work: _____________ and _______________
1. Construct one triangle that has lengths 4cm, 5cm, 7cm and another that has lengths 7cm, 7cm, and 4 cm
2. Construct a triangle that has leg lengths 5cm, 6cm, and an included angle of 50 degrees.
3. Construct a triangle that has leg lengths 4cm, 4cm and an included angle of 60 degrees. What kind of
triangle is this?
Name: _________________________
Period: ________
4.9 Proving Triangles Congruent
For each problem give the correct naming order of the congruent triangles. Write that name in order on the
lines for the problem number (see box at bottom). Also, indicate which postulate or theorem is being used.
1.
B
I
C
A
W
J
R
H
G
A
7.
A
C
B
B
P
K
ABC   _______ by _________
E
3.
C
T
G
A
B
N
ABC   _______ by _________
T
S
A
D
6.
A
E
B
C
H
Y
ABC   _______ by _________
ABC   _______ by _________
8.
9.
F
S
E
D
L
G
A
5.
H
ABC   _______ by _________
10. C
C
C
E
I
H
S
ABC   _______ by _________
GHJ   _______ by _________
B
A
B
R
ABC   _______ by _________
4.
2.
N
A
B
Y
C
ABC   _______ by _________
H
JKL   _______ by _________
E
D
A
T
L
K
DEF   _______ by _________
11.
J
A
12.
O
M
K
N
S
A
MNO   _______ by _________
___ ___ ___ ___ ___ _O_ ___ ___ _N_ ___ ___ ___ _S_ ___ ___ _E_ ___ _I_ ___ ___ ___ ___ ___ _T_ ___
4 4
4 8
8
8 12
12 12 2
2
2
5
5 5
9 9
9
6
___ ___ ___ _E_ _E_ ___ ___ ___ _O_ ___ ___ _N_ ___ _U_ ___ ___ ___ ___ _T_ ___ _E_ ___ ___ _I_ ___ .
6
6 10
10 10 1
1
1
3
3 3 7
7
7
11 11
11
(When you are done with the puzzle, there are: 3 SAS, 5 AAS, 2 ASA, and 2 SSS instances.)
13)
A) Translation
B) Vertical Reflection
C) Rotation
D) Horizontal Reflection
14)
There are five different ways to find triangles are congruent: SSS, SAS, ASA, AAS and HL.
For each pair of triangles, select the correct rule. Indicate if there isn’t enough information.
15) a) Mark the diagram with the given information.
b) Look for any other given information that could help show that the two triangles are congruent. Do they
overlap anywhere? Share any side or any angle? Mark it in the diagram.
c) You should have enough information to prove the triangles are congruent. Fill in the proof.
Statements
Reasons
1)
1) Given
2)
2) Given
3)
3)
4)
4)
d) What do you think is true about ∠𝐴 𝑎𝑛𝑑 ∠𝐶? Explain:
Name: _________________________
Period: ________
4.10 Proving Isosceles Conjectures
In this space, draw a large isosceles triangle. Use tools appropriately, do not freehand. Mark the two
congruent sides of the triangle. Precision is essential.
1) Label the vertices RED with R begin the vertex angle (included by the two congruent sides).
2) Carefully construct the angle bisector of ∠R. Label point Q on ̅̅̅̅
𝐷𝐸 where the angle bisector ray
̅̅̅̅ . Mark the congruent angles.
intersects 𝐷𝐸
3) Complete the proof to show that Δ ERQ ≅ Δ DRQ
Statement
1. ___________________________
2. ∠ERQ ≅ ∠ DRQ
3. ___________________________
4. Δ ERQ ≅ Δ DRQ
Reason
1. Given
2. ___________________________
3. ___________________________
4. SAS
4) Using CPCTC, what other parts of the triangle are congruent?
________ ≅ ________ and ________ ≅ ________
5) On your diagram, measure 𝐸𝑄 𝑎𝑛𝑑 𝐷𝑄, what can you say about point Q?
6) On your diagram, measure ∠𝐸𝑄𝑅 𝑎𝑛𝑑 ∠𝐷𝑄𝑅, what can you say about these angles?
7) Describe a series of transformations that would map Δ ERQ onto Δ DRQ. Be specific.
The isosceles triangle theorems are frequently abbreviated as
1. ΔXYZ is an isosceles triangle with perimeter 48 cm
If XY = 18
Find XW _______
Δ MPQ is an isosceles triangle. 𝑚∠𝑃𝑄𝑀 = 45° 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚∠𝑄𝑃𝑁
2. M
N
P
3.
Q
5
𝑥
2
2x
−5
Find AB
2x
(3y – 5)0 Solve for y
4.
400
5.
(5x + 15)0 solve for x
Name: _________________________
Period: ________
4.11 CPCTC worksheet
MARK THE DIAGRAMS WITH THE GIVEN INFORMATION!
#1: HEY is congruent to MAN by ______.
What other parts of the triangles are congruent by CPCTC?
______  ______
______  ______
H
______  ______
M
A
Y
E
N
#2:
CAT  ______, by _____
C
R
THEREFORE:
______  ______, by CPCTC
______  ______, by CPCTC
______  ______, by CPCTC
A
T
P
#3:
C
Given:
AC  AR and 1  2
Prove:
3  4
R
3
A
1
L
Proof:
1.
2.
3.
4.
5.
AC  AR
____________
CAL  RAS
LCA  SRA
3  4
1.
2.
3.
4.
5.
_______________
Given
________________
________________
________________
4
2
S
MARK THE DIAGRAMS WITH THE GIVEN INFORMATION!
#4:
L
Given:
Prove:
M
NLM  LNO and OLN  MNL
M  O
O
N
Proof:
1.
2.
3.
4.
5.
NLM  LNO
1.
2.
3.
4.
5.
_________________
_________________
LMN  ______
_________________
_________________
Given
Reflexive Property of 
_________________
_________________
C
#5
Given: AC  BC and AX  BX
Prove: 1  2
Proof:
1. __________________________
2. __________________________
3. AXC  _______
4. ________________
1.
2.
3.
4.
1
2
3
4
A
X
B
Given
Reflexive Prop. of Congruence
____________
____________
#6
W
Z
Given: 1  2 and 3  4
Prove: XY  ZW
Proof:
1. __________________________
2. XZ  XZ
3. XWZ  _______
4. ________________
2
4
1
X
1.
2.
3.
4.
3
Given
________________
________________
________________
Y
Name: ______________________________Period: __________
2
3.
4.
4.12 Using Proof Blocks
5.
6.
7.
8.
Name: ______________________________
Period: __________
1. Given: 𝐵𝐶 ≅ 𝐷𝐸 & ∠𝐵 ≅ ∠𝐸
Prove: 𝐴𝐶 ≅ 𝐴𝐷
2. Given: ∠𝐷 ≅ ∠𝑃, ∠𝐸 ≅ ∠𝑄, 𝐸𝐷 ≅ 𝑃𝑄
Prove: 𝐷𝐹 ≅ 𝑃𝑅
3. Given that ∠𝐺 ≅ ∠𝐾, and the information in the diagram,
prove 𝐻𝐼 ≅ 𝐽𝐿
4.13Proof blocks & CPCTC
4. Using the information in the diagram prove that ∠𝑀 ≅ ∠𝑂.
̅
5. Given that 𝐺𝐻 ∥ 𝐽𝐼, I is the midpoint of 𝐻𝐾 and ̅̅̅̅
𝐺𝐻 ≅ 𝐽𝐼
Prove: ∠𝐺 ≅ ∠𝐽
6. Given that 𝑀𝑁 ⊥ 𝑂𝑃, and the information in the diagram to prove
that 𝑂𝑃 is the angle bisector of ∠𝑀𝑃𝑁
Name: _________________________________Period: _________
1) Classify ∆ABC by its angles and its side lengths ___________
4.14 Quiz 4 Review
_________________
2) Classify each triangle by its side length
∆ABD ______________ and ∆ADC ___________________
3) While surveying a triangular plot of land, a surveyor finds 𝑚∠𝑆 = 430 .
The measure of ∠𝑅𝑇𝑃 is twice that of ∠𝑅𝑇𝑆. What is the 𝑚∠𝑅?
Given ∆XYZ ≅ ∆JKL, identify the congruent corresponding parts
̅ ≅ _____
4) 𝐽𝐿
5) ∠𝑌 ≅ ____
6) ∠𝐿 ≅ ____
̅̅̅̅
̅̅̅̅ 𝑎𝑛𝑑 𝑆𝑄
8) Given: T is the midpoint of both 𝑃𝑅
Prove: ∆PTS ≅ ∆RTQ
9 & 10: Find the Measures of each angle
Statements
7) ̅̅̅̅
𝑌𝑍 ≅ ___
Reasons
11. Find the measure of ∠𝐴𝑈𝑇
13.
12. Find the measure of ∠CBD
14.
15. Look at the diagrams below and determine which (if any) triangle congruence theorems you can use to
prove two triangles are congruent. Circle the ones that work and label them with SSS, SAS, AAS, ASA or HL.