Lesson 3-1

Lesson 3-1: Triangle Fundamentals

1

Lesson 3-1

Triangle Fundamentals

Polygon


2

Polygon - closed figure, in a plane (2-D), made of segments intersecting only at their endpoints

EX)

NOT EX)

Triangles Triangle- sided polygon- ABC


3

3

Sides of a -

ABBCAC

Vertices-

ABC

A

B C


4

Naming Triangles

For example, we can call the following triangle:

Triangles are named by using its vertices.

∆ABC ∆BAC

∆CAB ∆CBA∆BCA

∆ACB

A

B

C


5

Opposite Sides and Angles

A

B C

Opposite Sides:

Side opposite to A :

Side opposite to B :

Side opposite to C :

Opposite Angles:

Angle opposite to : A

Angle opposite to : B

Angle opposite to : C

BC

AC

AB

BC

AC

AB


6

Classifying Triangles by Angles

Acute:

Obtuse:

A triangle in which all angles are less than 90˚.

A triangle in which and only angle is greater than 90˚& less than 180˚

100

45

35 B

C

A

50 60

70

G

H I

3

1 1


7

Classifying Triangles by Angles

Right:

Equiangular:

A triangle in which and only angle is 90˚

A triangle in which all angles are the same measure.

34

56

90B C

A

60

6060C

B

A

1 1


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Classifying Triangles by Sides

Equilateral:

Scalene:No 2 sides are congruent

A triangle in which all 3 sides are different lengths.

Isosceles: A triangle in which at least 2 sides are equal.

A triangle in which all 3 sides are equal.

AB

= 3

.02

cm

AC

= 3.15 cm

BC = 3.55 cm

A

B CAB =

3.47

cmAC = 3.47 cm

BC = 5.16 cmBC

A

HI = 3.70 cm

G

H I

GH = 3.70 cm

GI = 3.70 cm


9

polygons

Classification by Sides with Flow Charts & Venn Diagrams

triangles

Scalene

Equilateral

Isosceles

Triangle

Polygon

scalene

isosceles

equilateral


10

polygons

Classification by Angles with Flow Charts & Venn Diagrams

triangles

Right

Equiangular

Acute

Triangle

Polygon

right

acute

equiangular

Obtuse

obtuse

Parts of a right


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LEG

LEG

HYPOTENUSE

Parts of an Isoceles


12

A

B C

LEG LEG

BASE

Base Angles

Vertex AngleThe congruent sides are called legs and the third side is called the base


13

Theorems & Corollaries

The sum of the interior angles in a triangle is 180˚.

Angle Sum Theorem:

Third Angle Theorem:If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Corollary 1: Each angle in an equiangular triangle is 60˚.

Corollary 2: Acute angles in a right triangle are complementary.

Corollary 3: There can be at most one right or obtuse angle in a triangle.

Auxillary Line: A line added to a picture to help prove something

Lesson 3-2: Isosceles Triangle 14

Isosceles Triangle Theorems

By the Isosceles Triangle Theorem,the third angle must also be x.Therefore, x + x + 50 = 180

2x + 50 = 1802x = 130x = 65

Example:

x

50

Find the value of x.

A

B C

, .If AB AC then B C

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Lesson 3-2: Isosceles Triangle 15

Isosceles Triangle TheoremsIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Example: Find the value of x. Since two angles are congruent, the sides opposite these angles must be congruent.

3x – 7 = x + 152x = 22X = 11

A

B C

50 50

3x - 7x+15

A

B C

, .If B C then AB AC


16

Exterior Angle Theorem

The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Exterior AngleRemote Interior Angles A

BC

D

m ACD m A m B

Example:

(3x-22)x80

B

A DC

Find the mA.

3x - 22 = x + 80

3x – x = 80 + 22

2x = 102

mA = x = 51°

Lesson 4-2: Congruent Triangles 17

Lesson 4-2

Congruent Triangles


Congruent Figures

Congruent figures are two figures that have the same size and shape.

IF two figures are congruent THEN they have the same size and shape.

IF two figures have the same size and shape THEN they are congruent.

Two figures have the same size and shape IFF they are congruent.

Congruent Triangles


ZY

MN ___

NR ___

MR ___

N

M

R

F E

D

≡ ≡=

=

│ │

∆MNR ______

M ____ D

N _____

R ______

E

F

∆ DEF

∆MNR ∆FED

Note:DE

EFDF


Congruent Triangles - CPCTC

If ABC PQR

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Two triangles are congruent IFF their corresponding parts (angles and sides) are congruent.

BC

A

QR

PA ↔ P; B ↔ Q; C ↔ R

Vertices of the 2 triangles correspond in the same order as the triangles are named.

Corresponding sides and angles of the two congruent triangles:

A

A P B Q C

B PQ BC Q C P

R

R A R

≡

≡

=

=

│

│


When referring to congruent triangles (or polygons), we must name corresponding vertices in the same order.

R

AY

S

UN

S

U

N

R

A

YSUN RAY

Also NUS YAR

Also USN ARY

Example…………


Example ………

M

O

N

TA SR

UP

E

1. Pentagon MONTA Pentagon PERSU

2. Pentagon ATNOM Pentagon USREP

3. Etc.

If these polygons are congruent, how do you name them ?

Lesson 4-3: SSS, SAS, ASA 23

Included Angles & Sides

& .A is the included angle for AB AC

& .B is the included angle for BA BC

& .C is the included angle for CA CB

A

B C

Included Angle:

Included Side:& .AB is the included side for A B

& .BC is the included side for B C

& .AC is the included side for A C

** *


Lesson 4-3

Proving Triangles Congruent(SSS, SAS, ASA)


PostulatesASA If two angles and the included side of one triangle are

congruent to the two angles and the included side of another triangle, then the triangles are congruent.

SAS If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

A

B C

D

E F

A

B C

D

E F


PostulatesSSS If the sides of one triangle are congruent to the sides of a

second triangle, then the triangles are congruent.

Included Angle: In a triangle, the angle formed by two sides is the included angle for the two sides.

Included Side: The side of a triangle that forms a side of two given angles.

A

B C

D

E F


Steps for Proving Triangles Congruent

1. Mark the Given.

2. Mark … Reflexive Sides / Vertical Angles

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts … in the order of the method.

5. Fill in the Reasons … why you marked the parts.

6. Is there more?


Problem 1 Given: AB CD BC DAProve: ABC CDA

Statements Reasons

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

A B

D C

SSS

Given

Given

Reflexive Property

SSS Postulate4. ABC CDA

1. AB CD2. BC DA3. AC CA


Problem 2 Step 1: Mark the Given Step 2: Mark vertical anglesStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

SAS

Given

Given

Vertical Angles.

SAS Postulate

: ;

Pr :

Given AB CB EB DB

ove ABE CBD

E

C

D

AB

1. AB CB2. ABE CBD

3. EB DB4. ABE CBD

Statements Reasons


Problem 3

Statements Reasons

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

ASA

Given

Given

Reflexive Postulate

ASA Postulate

: ;

Pr :

Given XWY ZWY XYW ZYW

ove WXY WZY

Z

W Y

X 1. XWY ZWY

2. WY WY3. XYW ZYW

4. WXY WZY

Lesson 4-4: AAS & HL Postulate 31

PostulatesAAS If two angles and a non included side of one triangle are

congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

HL If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

A

B C

D

E F

A

B C

D

E F


Problem 1

Statements Reasons

Step 1: Mark the Given Step 2: Mark vertical anglesStep 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

AAS

Given

Given

Vertical Angle Thm

AAS Postulate

Given: A C BE BDProve: ABE CBD

E

C

D

AB

1. A C2. ABE CBD

3. BE BD

4. ABE CBD


Problem 2

3. AC AC2. AB AD

1. ,ABC ADCright s

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

HL

Given

Given

Reflexive Property

HL Postulate

Given: ABC, ADC right s AB ADProve: ABC ADC

CB D

A

4. ABC ADC

Statements Reasons

Documents

Lesson 3-1