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2.2b: Exploring Congruent
Triangles with Proofs
CCSS
G-CO.7
Use the definition of congruence in terms of rigid motions to show that two triangles
are congruent if and only if corresponding pairs of sides and corresponding pairs of
angles are congruent
G-CO.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of rigid motions.
A proof is an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true.
An important part of writing a proof is giving
justifications to show that every step is valid.
Example 1: Solving an Equation in Algebra
Solve the equation 4m – 8 = –12. Write a
justification for each step.
4m – 8 = –12
+8
+8
Given equation
Addition Property of Equality
4m
Simplify.
= –4
Division Property of Equality
m = –1
Simplify.
Check It Out! Example 1
Solve the equation
for each step.
. Write a justification
Given equation
Multiplication Property of Equality.
t = –14
Simplify.
Example 3: Solving an Equation in Geometry
Write a justification for each step.
NO = NM + MO
Segment Addition Post.
4x – 4 = 2x + (3x – 9) Substitution Property of Equality
4x – 4 = 5x – 9
–4 = x – 9
5=x
Simplify.
Subtraction Property of Equality
Addition Property of Equality
Check It Out! Example 3
Write a justification for each step.
mABC = mABD + mDBC
8x° = (3x + 5)° + (6x – 16)°
8x = 9x – 11
–x = –11
x = 11
 Add. Post.
Subst. Prop. of Equality
Simplify.
Subtr. Prop. of Equality.
Mult. Prop. of Equality.
Remember!
Numbers are equal (=) and figures are congruent
().
Example 4: Identifying Property of Equality and
Congruence
Identify the property that justifies each
statement.
A. QRS  QRS
Reflex. Prop. of .
B. m1 = m2 so m2 = m1
Symm. Prop. of =
C. AB  CD and CD  EF, so AB  EF. Trans. Prop of 
D. 32° = 32°
Reflex. Prop. of =
Check It Out! Example 4
Identify the property that justifies each
statement.
4a. DE = GH, so GH = DE. Sym. Prop. of =
4b. 94° = 94° Reflex. Prop. of =
4c. 0 = a, and a = x. So 0 = x. Trans. Prop. of =
4d. A  Y, so Y  A
Sym. Prop. of 
Ex.
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z. Trans. Prop. of =
4. DEF  DEF
Reflex. Prop. of 
5. AB  CD, so CD  AB.
Sym. Prop. of 
2-Column Proof
• Numbered statements and corresponding
reasons in a logical order organized into 2
columns.
statements
reasons
1.
1.
2.
2.
3.
3.
etc.
2-Column Proof (Con’t)
statements
reasons
You can only use:
• Given
•Definitions
•Postulates
•Theorems
•Properties
Ex: Given: PQ=2x+5
QR=6x-15
PR=46
Prove: x=7
1.
2.
3.
4.
5.
6.
Statements
PQ=2x+5, QR=6x-15,
PR=46.
PQ+QR=PR
2x+5+6x-15=46
8x-10=46
8x=56
x=7
Reasons
1. Given
2.
3.
4.
5.
6.
Seg Add Post.
Subst. prop of =
Simplify
Add prop of =
Division prop of =
Ex: Given: Q is the midpoint of PR.
PR
Prove: PQ and QR =
2
1.
2.
3.
4.
5.
6.
Statements
Q is midpt of PR
PQ=QR
PQ+QR=PR
QR+QR=PR
2QR=PR
QR= PR
2
PR
7. PQ=
2
1.
2.
3.
4.
5.
6.
Reasons
Given
Defn. of midpt
Seg Add. post
Subst. prop of =
Simplify
Division prop of =
7. Subst. prop
H
Example–
A
W
Given: HA || KS
AW WK
Prove: HAW  SKW
K
S
Reasons
Statement
HA || KS, AW WK
1
Given
2
HAW SKW
2
Alt. Int. Angles are congruent
3
HWA SWK
3 Vertical
4
HAW  SKW
1
4
Angles are congruent
ASA Postulate
Example
AMT is isosceles with vertex MAT
Given:
MAT is bisected by AH
Prove: MH  HT
Statement
1)
Reason
AMT is isosceles with vertex MAT 1) Given
MAT is bisected by AH
Example
Given:
AW || TB
E is the midpoint of WB
Prove: AW  TB
Statement
1) AW || TB
E is the midpoint of WB
2)
Reason
1)
2)
Given
Example
AB  CB
Given:
BDA and BDC are right
ABD and CBD are right tria ngles
Prove: A  C
Statement
1) AB  CB
BDA and BDC are right
ABD and CBD are right tria ngles
Reason
1)
2)
2)
Given
Given:
Prove:
D is the midpoint of RY
DYF  DRA
FD  AR
Reasons
Statement
1 D ,is the midpoint of RY
DYF  DRA
1
Given