Name: Period: 4.1 Tricky Triangles 1) Triangle Sum ...
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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 d= ______
380 120
e e= ______ h= ______
f g 1240 380 f= ______ i= ______
h g= ______ j = ______
i
1380 450
750
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 5
line n
a) Given: m || n
Prove: The sum of the angles of ΞABC is 1800
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 πβ 1 = πβ 2 +πβ 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 βnot possible.β
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: ________ 4.3 Overlapping Triangles
For 1 & 2, shade a different triangle in each image.
1)
2)
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
π₯ + π¦ > π§
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, 7
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
βneitherβ
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 π»πΌ β πΎπΌ and from the diagram there
is a pair of vertical angles so β π»πΌπ½ β β πΎπΌπ, so we need
One more piece of needed information: ___________________
To prove βπ»πΌπ½ β βπΎπΌπ ππ¦ ππ΄π
9. prove by: SSS
One more piece of needed information: ___________________
Ξ KMH β Ξ ________
10. C A
prove by: SAS
T F One more piece of needed information: ___________________
Ξ CAT β Ξ ________
11. prove by: SAS
One more piece of needed information: ___________________
Ξ MKL β Ξ ________
12. prove by: SSS
One more piece of needed information: ___________________
Ξ XZY β Ξ ________
Β©a O2W0G1j5d aKZu\tcaW cS\oWfHtowXaorHeg LLjLpCU.n e ^ADlYl[ crdikgqhAtlsA arYeqsNeErGvveOdg.w i PMaa`dNeA Kwziytchu jIOnrfCignYivtZeV lGeeBoqmyeytJrSyX.
Worksheet by Kuta Software LLC
4.7 Geometry
ASA and AAS Congruence
Name___________________________________
Period____
Β©l t2_0e1z5z QK[uztjas zSto_fpt[wHaYr^eX KLMLYC\.u [ uAhlTlN orsixgYhztbss PrQelsFewrivjeEdX.
State if the two triangles are congruent. If they are, state how you know.
1) 2)
3) 4)
5) 6)
7) 8)
9) 10)
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 doesnβt 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 donβt 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. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
___ ___ ___ ___ ___ _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.)
W
A
B I
C R B
C G
A N B
A S
C E
G
J R
H E C
A E
B Y C
B S
A D
O
M K
N A
H T
A T H
C
B I
A L K
J T
L H D
F S
E N
H A A
B
C G
A K A
B Y
C D
P E S
ABC _______ by _________
ABC _______ by _________
ABC _______ by _________
ABC _______ by _________
GHJ _______ by _________
ABC _______ by _________
DEF _______ by _________
ABC _______ by _________
JKL _______ by _________
ABC _______ by _________
ABC _______ by _________
MNO _______ by _________
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 isnβt 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 β π΄ πππ β πΆ? 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 β R. Label point Q on π·πΈΜ Μ Μ Μ where the angle bisector ray
intersects π·πΈΜ Μ Μ Μ . Mark the congruent angles.
3) Complete the proof to show that Ξ ERQ β Ξ DRQ
Statement Reason
1. ___________________________ 1. Given
2. β ERQ β β DRQ 2. ___________________________
3. ___________________________ 3. ___________________________
4. Ξ ERQ β Ξ DRQ 4. SAS
4) Using CPCTC, what other parts of the triangle are congruent?
________ β ________ and ________ β ________
5) On your diagram, measure πΈπ πππ π·π, what can you say about point Q?
6) On your diagram, measure β πΈππ πππ β π·ππ , what can you say about these angles?
7) Describe a series of transformations that would map Ξ ERQ onto Ξ DRQ. Be specific.
The isosceles triangle theorems are frequently abbreviated as
1. ΞXYZ is an isosceles triangle with perimeter 48 cm
If XY = 18
Find XW _______
2. M Ξ MPQ is an isosceles triangle. πβ πππ = 45Β° ππππ π‘βπ πβ πππ
N
P Q
3. 2x 5
2π₯ β 5 Find AB
2x
4. (3y β 5)0 Solve for y
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?
______ ______
______ ______
______ ______
#2:
CAT ______, by _____
THEREFORE:
______ ______, by CPCTC
______ ______, by CPCTC
______ ______, by CPCTC
#3:
Given: ARAC and 21
Prove: 43
Proof:
1. ARAC
2. ____________
3. RASCAL
4. LCA SRA
5. 43
1. _______________
2. Given
3. ________________
4. ________________
5. ________________
M
A
N
Y
E
H
L
C
S
R
4 3
2 1
C
T P
A
R
A
MARK THE DIAGRAMS WITH THE GIVEN INFORMATION!
#4:
Given: LNONLM and MNLOLN
Prove: OM
Proof:
1. LNONLM
2. _________________
3. _________________
4. LMN ______
5. _________________
1. _________________
2. Given
3. Reflexive Property of
4. _________________
5. _________________
#5
Given: BCAC and BXAX
Prove: 1 2
Proof:
1. __________________________ 1. Given
2. __________________________ 2. Reflexive Prop. of Congruence
3. AXC _______ 3. ____________
4. ________________ 4. ____________
#6
Given: 1 2 and 3 4
Prove: ZWXY
Proof:
1. __________________________ 1. Given
2. XZXZ 2. ________________
3. XWZ _______ 3. ________________
4. ________________ 4. ________________
M
N O
L
C
X B A
1 2
4 3
W
X Y
Z
1
2 3
4
Name: ______________________________ Period: __________ 4.13Proof blocks & CPCTC
1. Given: π΅πΆ β π·πΈ & β π΅ β β πΈ
Prove: π΄πΆ β π΄π·
2. Given: β π· β β π, β πΈ β β π, πΈπ· β ππ
Prove: π·πΉ β ππ
3. Given that β πΊ β β πΎ, and the information in the diagram,
prove π»πΌ β π½πΏ
4. Using the information in the diagram prove that β π β β π.
5. Given that πΊπ» β₯ π½πΌ, I is the midpoint of π»πΎ and πΊπ»Μ Μ Μ Μ β π½οΏ½Μ οΏ½ Prove: β πΊ β β π½
6. Given that ππ β₯ ππ, and the information in the diagram to prove
that ππ is the angle bisector of β πππ
Name: _________________________________Period: _________ 4.14 Quiz 4 Review
1) Classify βABC by its angles and its side lengths ___________ _________________
2) Classify each triangle by its side length
βABD ______________ and βADC ___________________
3) While surveying a triangular plot of land, a surveyor finds πβ π = 430.
The measure of β π ππ is twice that of β π ππ. What is the πβ π ?
Given βXYZ β βJKL, identify the congruent corresponding parts
4) π½οΏ½Μ οΏ½ β _____ 5) β π β ____ 6) β πΏ β ____ 7) ππΜ Μ Μ Μ β ___
8) Given: T is the midpoint of both ππ Μ Μ Μ Μ πππ ππΜ Μ Μ Μ
Prove: βPTS β βRTQ Statements Reasons
9 & 10: Find the Measures of each angle
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