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Name ____________________________________ Date ___________________
10R Unit 6: Congruent Triangles
Period __________________
Day 1 Guided Notes – Congruence and Triangles
HW: Pg 222-223 #10-19, 30-31, 33, 40-41
Warm Up:
Figure is not drawn to scale.
a
70°
b
4y +14 °
If y = 23, what do we know about line a and b?
1.
a b
3.
ab
2.
a b
4.
we cannot determine their relationship
Vocabulary
Congruent Figures: In two congruent figures, all the parts of one figure are
_________________________ to the ___________________________ parts of the
other figure.
Corresponding Parts: In congruent polygons, the corresponding parts are the
corresponding _____________________________ and the corresponding
_______________.
Are these figures congruent? Remember, two figures are congruent if they have exactly
the same size and shape. In other words, they would fit perfectly one on top of the other!
Be careful, you may have to rotate or flip them around!!
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Examples:
1) Write a congruence statement for the triangles. Identify all pairs of congruent
corresponding parts.
To help you identify corresponding parts, turn the triangles around!
Corresponding Sides
Corresponding Angles
AB  _______
 A  _______
AC  _______
 B  _______
CB  _______
 C  _______
Therefore,  ABC  ___________________________________________________.
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2) Write a congruence statement for the triangles. Identify all pairs of congruent
corresponding parts.
Corresponding Sides
Corresponding Angles
AB  _______
 A  _______
BC  _______
 B  _______
AC  _______
 C  _______
Therefore,  ABC  ___________________________________________________.
Using Congruent Figures to find missing values.
1) In the diagram, QRST  WXYZ.
Therefore,
QR  _______
 Q  _______
RS  _______
 R  _______
ST  _______
 S  _______
QT  _______
 T  _______
a. Find the value of x
b. Find the value of y
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2) In the diagram, JFGH  VSTU.
Therefore,
JF  _______
 J  _______
FG  _______
 F  _______
GH  _______
 G  _______
JH  _______
 H  _______
a. Find the value of x
b. Find the m  G
Theorem 4.3 Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the
third angles are ___________________________________.
If  A   D and  B   E, then  C   F
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Examples:
1) Find m  V
2) Find the value of x.
3) In the diagram below, E is the midpoint of AC and BD. Show that ABE  CDE .
Corresponding Sides
Why?
AB  _______
BE  _______
AE  _______
Corresponding Angles
Why?
 A  _______
 B  _______
AEB  _______
Therefore, ABE  CDE because _____________________________________
________________________________________________________________.
4) In the diagram, what is the measure of  D?
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Day 2 Guided Notes – SSS
HW: Pg 230-231 #4, 9-10, 18
Warm Up:
1. ____________________________ 2. _____________________________
3. ___________________________
4. _____________________________
Side-Side-Side (SSS) Congruence Postulate: Proof Reason #_____
If three sides of one triangle are congruent to three sides of a second triangle, then the
two triangles are __________________________.
If Side AB  _____,
Side BC  _____ and
Side AC  _____ ,
Then
 ABC  _______.
Examples:
Decide if the congruence statement is true. Explain your reasoning.
1)
 JKL   MKL
2)  RST   TVW
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3)
 DFG has vertices D(-2, 4), F(4,4), and G(-2, 2).  LMN has vertices L(-3, -3),
M(-3, 3) and N(-1, -3). Graph the triangles in the same coordinate plane and show
they are congruent.
Proof 1:
Given: AB  BC
D is the midpoint of AC
Prove:  ABD   CBD
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Proof 2:
Given: QR  WT
s is the midpoint of RT
QS  WV
ST  TV
Prove:  QRS   WTV
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Proof 3:
Given: DC  BA
CE  AF
DF  BE
Prove:  AEB   CFD
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Day 3 Guided Notes – SAS and HL
HW: Pg 231-232 #11-14, 16-17, 24-26
Vocabulary
Leg of a Right Triangle: A side ____________________ to the right angle.
Hypotenuse: In a right triangle, the side _____________________ the right angle.
Side-Angle-Side (SAS) Congruence Postulate: Proof Reason # ______
If two sides and the _________ angle of one triangle are congruent to two sides and the
_________ angle of a second triangle, then the two triangles are __________________.
If Side
Angle
Side
RS  _____,
 R  _____ and
RT  _____ ,
Then  RST  _______.
In the diagram, ABCD is a rectangle. What can you conclude about  ABC and  CDA?
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Theorem 4.5: Hypotenuse-Leg Congruence Theorem: Proof Reason # ________
If the hypotenuse and a leg of a right triangle are __________________ to the
hypotenuse and a leg of a second triangle, then the two triangles are ___________.
USE WHEN YOU HAVE A RIGHT TRIANGLE AND TWO SIDES
AND NON-INCLUDED ANGLE…
NO SSA OR ASS ALLOWED!
If Hypotenuse BC  EF and
Leg
AB  DE
Then  ABC   DEF
Only works when two triangles are right angles…must first state that they are…
rt ∆ has 1 rt 
Proof Reason # ______
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Proof Practice: SAS
Given: AD  AB
AE bisects  DAB
Prove:
 ADC   ABC
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Proof Practice: HL
Given: AC  EC
AB  BD
ED  BD
AC is a bisector of BD
Prove:  ABC   EDC
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More Proof Practice…complete and come after school for extra help!
1) Given: AD  CB
AD ll CB
Prove:  ADB   CBD
2) Given: Line SYT and Line SXR
SX  SY
XR  YT
Prove:  RSY   TSX
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3) Given: DB  AC
DA  AB
CB  AB
Prove:  DAB   CBA
4) Given: DA  CB
AB  DA
DA ll BC
Prove:  DAB   CBA
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Day 4 Guided Notes – ASA and AAS
HW: Pg 238 #8-9, 11-12, 14-15
Angle-Side-Angle (ASA) Congruence Postulate: Proof Reason # ____
If two angles and the _____________ side of one triangle are congruent to two angles
and the _____________ side of a second triangle, then the two triangles are ________.
If Angle
Side
Angle
 A  _____,
AC  _____ and
 C  _____,
Then  ABC  _______.
Theorem 4.6: Angle-Angle-Side (AAS) Congruence Theorem: Proof Reason # _____
If two angles and a _________________ side of one triangle are congruent to two angles
and the corresponding __________________ side of a second triangle, then the two
triangles are _____________________.
If Angle
Angle
Side
 A  _____,
 C  _____ and
BC  _____
Then  ABC  _______.
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Proof Practice:
1) Given: DA bisects  BAC
AD  BC
Prove:  ABD   ACD
2) Given:
 BAC   BCA
1  2
Prove:  ADC   CEA
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3) Given:
1  2
DF  CE
 A and  B are right  s
1
2
Prove:  AFD   BEC
4) Given: SU ll AT
UA bisects ST at N
Prove:  SUN   TAN
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Day 5 Guided Notes – CPCTC
HW: Finish Notes/Study for Proof Quiz (SSS, SAS and HL)
By definition, congruent triangles have congruent corresponding parts. So, if you can
prove that two triangles are congruent, you know that their corresponding parts must
be congruent as well.
CPCTC Proof Reason # _____
Corresponding Parts of Congruent Triangles are Congruent!
Review: How to do a proof:
1. Write givens.
2. Mark up your picture.
3. Make conclusions from your givens ONLY.
4. Use ALL your givens, check for vertical angles, reflexive, linear pair.
5. Look for SSS, SAS, AAS, ASA, HL pattern to prove the triangles are congruent.
6. State other corresponding parts are congruent due to CPCTC.
CPCTC Proofs!
1) Given: AC  BC
D is the midpoint of AB
Prove:  ACD   BCD
Hint: First prove  ACD   BCD by SSS… then use CPCTC!
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2) Given:  1   2
AB  DE
Prove: DC  AC
Hint: First prove  ABC   DEC by AAS…
then use CPCTC!
3) Given:  1   2
3 4
Prove:  ABD   ACD
Hint: First prove  BDE   CDE by AAS… then use CPCTC!
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4) Given: GH  KJ, FG  LK,
 FJG and  LHK are rt  s
Prove:  FJK   LHG
Hint: First prove  FJG   LHK by HL (rt  ) … then use CPCTC!
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Day 6 Guided Notes – Isosceles & Equilateral Triangles
HW: Pg 253 #2–4, 6-12, 16-19, 25
1. Proof Reason # ________:
isos ∆ ↔ 2  sides or isos ∆ ↔ 2  base s
(depending on what you are given or what you need to prove)
2. Proof Reason # _______ :
equil ∆ ↔ 3  sides or equil ∆ ↔ 3  s
(depending on what you are given or what you need to prove)
Vocabulary
Legs: The legs of an isosceles triangle are the two congruent sides.
Vertex Angle: The angle formed by the legs.
Base: The side that is not the leg.
Base Angles: The two angles adjacent to the base.
Label the legs, vertex angle, base and base angles in the
isosceles triangle.
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Base Angles Theorem: Proof Reason #_____
If two sides of a triangle are congruent, then the angles ____________ them are
___________________.
If AB  AC, then  B   C
Converse of Base Angles Theorem: Proof Reason #_____
If two angles of a triangle are congruent, then the _______ opposite them are congruent.
If  B   C, then AB  AC
Let’s Prove It!
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Examples involving isoscesles and equilateral triangles.
1) Find the measures of  R,  S, and  T.
2) Find the values of x and y in the diagram.
3) Complete the statement:
If FH  FJ, then  _________   _________
4) Complete the statement:
If  FGK is equiangular and FG = 15, then GK = _____
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5) Find the values of x and y.
6) Find the value of x.
7) The measure of a base angle of an isosceles triangle is 13 degrees more than
three times the vertex angle. Find the measure of the vertex angle.
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8) Use the diagram to complete the statements.
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Day 7 Guided Notes – Congruence Transformations
HW: Pg 582-583 #6-11
Warm Up: Look at the congruent triangles.
1. Draw line segments to match corresponding sides.
AB
BC
AC
DE
DF
EF
2. Draw line segments to match corresponding angles.
A
B
C
E
D
F
Two figures are congruent if and only if there is a sequence of one or more rigid
motions (isometries) that maps one figure onto another.
Example: If a triangle is transformed by the composition of a reflection and a
translation, the image is congruent to the given triangle.
1. List the four isometries:
____________________________________________________________________________________
2. The composition of two or more isometries is an ____________________________________
3. Circle the transformations that make up a glide reflection.
translation
reflection
rotation
dilation
4. When a figure is transformed by an isometry, what is true about the corresponding
angles and the corresponding sides of the image and preimage? ________________.
5. Is the composition of a rotation followed by a glide reflection a congruence
transformation? Explain.
_________________________________________________________________
_________________________________________________________________
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The composition (Rt
T  2,3 ) (  ABC) =  XYZ.
1. Compositions of isometries preserve angle measure, so corresponding angles have
equal measures. Fill in the blanks to make true statements.
2. By definition, isometries preserve distance. So, corresponding
side lengths have equal measures. Draw line segments to match
corresponding side lengths.
AB
BC
AC
YZ
XZ
XY
Which pairs of figures in the grid are congruent? For each pair, what is a sequence
of rigid motions that map one figure to the other?
Circle the correct words to complete each sentence:
1. To map parallelogram ABCD onto parallelogram HIJK,
translate ABCD 2 / 5 / 6 units to the right / left and
3/ 5 / 6 units up / down.
2. To map  UVW onto triangle  QNM, reflect
 UVW across y = -1 / 0 / 1. Then translate  UVW 6 units
right / left / up / down.
Remember … if there is a sequence of one or more rigid motions (i.e. isometries) that
maps one figure onto another then two figures are congruent!
Fill in the blanks:
3.
4.
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What is a congruence transformation
that maps  NAV to  BCY?
Fill in the blanks to complete the statements.
More Practice:
1) In the diagram below,  JQV ≅  EWT. What is the congruence
transformation that maps  JQV to  EWT?
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2) Which pairs of figures in the grid are congruent? For each pair, what is a
sequence of rigid motions that maps one figure to the other?
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Common Core Review Questions
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3)
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