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Properties of Congruent Segments and Triangles Reflexive Property AB AB and ABC ABC Symmetric Property If AB CD, then CD AB ABC DEF then DEF ABC Transitive Property If AB CD and CD EF, then AB EF If ABC DEF, and DEF JKL, then ABC JKL
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Given: Triangle ABC Triangle DEF and <C <F•Solve for x and find the measure of the acute angles in the right triangles.•What triangle theorems does this solution use?
A˚
(4x – 8 )˚
Warm Up
B˚
C˚ D˚E˚
F˚
(x – 7 )˚
Proving Theorems about Triangles
• Theorems are true statements that follows as a result of other true statements
• A two-column proof has numbered statements and reasons that show the logical order of an argument
• A paragraph proof is a proof that has the same information as a two-column proof; but is written in a paragraph
Properties of Congruent Segments and Triangles
• Reflexive PropertyABAB and ABC ABC
• Symmetric PropertyIf ABCD, then CDAB ABC DEF then DEF ABC
• Transitive PropertyIf ABCD and CDEF, then ABEFIf ABC DEF, and DEF JKL, then ABC JKL
Properties of Congruent Segments and Triangles
• Substitution PropertyIf a = b than a can be substituted for b in an equation or expressionIf AB=CD, then AB can be substituted for CD
Lesson 2.2Lesson 2.2 Proving Triangles Congruent Proving Triangles Congruent
Example 1Example 1 State the PropertyState the Property
[a] [b]
A B
C
D E
F A
B
C
D
E
Example 2 AExample 2 A Triangle ProofsTriangle Proofs
Given: See Diagram
Prove: A
B
C
D
EDCABC
Statements ReasonsE
12
Example 2 BExample 2 B Triangle ProofsTriangle Proofs
Given: ABCD is a Rectangle
Prove: CDAABC
Statements Reasons
A
B C
D
Triangle Proofs Part I WorksheetTriangle Proofs Part I Worksheet
Warm-Up (2.2)Warm-Up (2.2)Given: Statements Reasons
1. 1.Prove:
2. 2.
3. 3.
4. 4.
A B
C
DE
DCACDEAB ,//DCACDEAB ,// GivenGiven
DCEACB VA =VA =
EB AIA =AIA =
DCEACB AASAAS
DCEACB
Triangle Proof Review WorksheetTriangle Proof Review Worksheet
Lesson 2.3Lesson 2.3 Proving Triangles Congruent & CPCTC Proving Triangles Congruent & CPCTC
Example 1Example 1 State PropertiesState Properties
[a] [b]U W
X
YZ
A B
CD
Example 2 AExample 2 A CPCTCCPCTC
Given: See Diagram
Prove: A
B
C
D
DB
Statements ReasonsE
12
1.
2.
3.
4.
1.
2.
3.
4.
DCBCECAC , GivenGiven
21 VA =VA =
EDCABC SASSAS
DB CPCTCCPCTC
Example 2 BExample 2 B
Given:
Prove:
Statements Reasons
A
B C
D
Triangle Proofs Part II WorksheetTriangle Proofs Part II Worksheet
1
2
CBADCBAD ,//
DB
CBADCBAD ,//21
ACAC
CDAABC
DB
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
GivenGiven
AIA =AIA =
Reflex.Reflex.
SASSAS
CPCTCCPCTC
Warm-UpWarm-Up
Use the following order pairs: A(2, 4) and B(-2, -6)
[1] Find the slope AB
[2] Find the slope // and | to AB
[3] Find the length of AB (simplify the radical)
Math I Skill Review Solving Basic Quadratic Equations
Step for Solving (Factoring Method)
[1] Set equation equal to zero
[2] Factor the non-zero side
[3] Identify the zeros of each factor (zero product property –take the opposite value)
Examples Worksheet 2.6 Examples Worksheet 2.6
Example 1 Factoring Method
[A] m2 – m – 6 = 0(m + 3)(m – 2) = 0(m + 3)(m – 2) = 0
{–3, 2}{–3, 2}
[B] m2 – 9m + 20 = 0(m – 4)(m – 5) = 0(m – 4)(m – 5) = 0
{4, 5}{4, 5}
Example 1 Factoring Method
[C] x2 + 5x – 36 = 0(x + 4)(x – 9) = 0(x + 4)(x – 9) = 0
{– 4, 9}{– 4, 9}
[D] n2 + 18n + 45 = 0(n + 3)(n + 15) = 0(n + 3)(n + 15) = 0
{–3, –5}{–3, –5}
Example 1 Factoring Method
[E] x2 = 12x – 20
(x – 2)(x – 10) = 0(x – 2)(x – 10) = 0
{2, 10}{2, 10}
[F] n2 – 100 = 48n
(n + 2)(n – 50) = 0(n + 2)(n – 50) = 0
{–2, 50}{–2, 50}
xx22 – 12x + 20 = 0 – 12x + 20 = 0 nn22 – 48x – 100 = 0 – 48x – 100 = 0
Examples Worksheet 2.6 Examples Worksheet 2.6