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