Download Ch 4 Triangles, Triangle Congruence

Lesson 3-1
Triangle
Fundamentals
Lesson 3-1: Triangle
Fundamentals
1
Polygon
Polygon - closed figure, in a plane (2-D), made
of segments intersecting only at their endpoints
EX)
NOT EX)
Lesson 3-1: Triangle
Fundamentals
2
Triangles

Triangle- 3 sided polygon-
Sides of a
ABC
Vertices-
A
A
B
C
AB
BC
AC
B
C
Lesson 3-1: Triangle
Fundamentals
3
Naming Triangles
Triangles are named by using its vertices.
For example, we can call the following triangle:
∆ABC
∆ACB
∆BAC
∆BCA
∆CAB
∆CBA
B
C
A
Lesson 3-1: Triangle
Fundamentals
4
Opposite Sides and Angles
Opposite Sides:
A
Side opposite to B : AC
Side opposite to A : BC
Side opposite to C : AB
B
C
Opposite Angles:
Angle opposite to BC : A
Angle opposite to AC : B
Angle opposite to AB : C
Lesson 3-1: Triangle
Fundamentals
5
Classifying Triangles by Angles
Acute: A triangle in which all
3 angles are less than 90˚.G
70
50
60
H
Obtuse:
1
I
A
1
A triangle in which and only angle is
greater than 90˚& less than 180˚
45
35 100 C
B
Lesson 3-1: Triangle
Fundamentals
6
Classifying Triangles by Angles
1
1
Right: A triangle in which and only angle is 90˚
A
56
B
90
34
C
Equiangular: A triangle in which all angles are the same measure.
B
60
A
60
Lesson 3-1: Triangle
Fundamentals
60
C
7
Classifying Triangles by Sides
No 2 sides are congruent
Scalene: A triangle in which all 3 sides are different lengths.
A
A
B
C
B
BC = 3.55 cm
C
BC = 5.16 cm
Isosceles: A triangle in which at least 2 sides are equal.
G
Equilateral: A triangle in which all 3 sides are equal.
GH = 3.70 cm
H
Lesson 3-1: Triangle
Fundamentals
HI = 3.70 cm
8
I
Classification by Sides
with Flow Charts & Venn Diagrams
polygons
Polygon
triangles
Triangle
scalene
Scalene
Isosceles
isosceles
equilateral
Equilateral
Lesson 3-1: Triangle
Fundamentals
9
Classification by Angles
with Flow Charts & Venn Diagrams
Polygon
polygons
triangles
Triangle
right
acute
Right
Obtuse
Acute
Equiangular
Lesson 3-1: Triangle
Fundamentals
equiangular
obtuse
10
Parts of a right
HYPOTENUSE
LEG
LEG
Lesson 3-1: Triangle
Fundamentals
11
Parts of an Isoceles
A
Vertex Angle
The congruent sides are
called legs and the third
side is called the base
LEG
LEG
Base Angles
B
BASE
Lesson 3-1: Triangle
Fundamentals
C
12
Theorems & Corollaries
Angle Sum Theorem:
The sum of the interior angles in a
triangle is 180˚.
A line added to a picture to help prove
Auxillary Line:
something
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.
Lesson 3-1: Triangle
Fundamentals
13
Isosceles Triangle Theorems
If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
A
If AB  AC , then B  C.
B
C
Example: Find the value of x. By the Isosceles Triangle Theorem,
the third angle must also be x.
Therefore, x + x + 50 = 180
50
2x + 50 = 180
2x = 130
x
x = 65
Lesson 3-2: Isosceles Triangle
14
Isosceles Triangle Theorems
If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
A
If B  C , then AB  AC.
B
C
Example: Find the value of x. Since two angles are congruent, the
A
sides opposite these angles must be
congruent.
3x - 7
x+15
3x – 7 = x + 15
2x = 22
50  C
B 50 
X = 11
Lesson 3-2: Isosceles Triangle
15
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.
Remote Interior Angles
Exterior Angle
mACD  mA  mB
Example: Find the mA.
B
x
A
(3x-22)
D
C
D
B
3x - 22 = x + 80
80
A
3x – x = 80 + 22
C
mA = x = 51°
2x = 102
Lesson 3-1: Triangle
Fundamentals
16
Lesson 4-2
Congruent
Triangles
Lesson 4-2: Congruent Triangles
17
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.
Lesson 4-2: Congruent Triangles
18
Congruent Triangles
N
R



D
____

E
_____

F
______
R
D
≡ ≡
M
F
ZY
=
DE
MN  ___
NR  ___
EF
DF
MR  ___
N
=
M 
E
Note:
∆MNR
DEF
 ∆______
∆MNR

Lesson 4-2: Congruent Triangles
∆FED
19
Congruent Triangles - CPCTC
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Two triangles are congruent IFF their corresponding parts
(angles and sides) are congruent.
A
If
ABC  PQR
A ↔ P; B ↔ Q; C ↔ R
B
C
≡
Vertices of the 2 triangles correspond in the same order
as the triangles are named.
P
Corresponding sides and angles of the two congruent triangles:
AB  PQ
B  Q
BC  QR
C  R
AC  PR
Lesson 4-2: Congruent Triangles
Q
≡
A  P
20
R
Example…………
When referring to congruent triangles (or polygons), we must
name corresponding vertices in the same order.
R
Y
S
R
U
A
N
Y
SUN  RAY
A
N
U
Also NUS  YAR
Also USN  ARY
S
Lesson 4-2: Congruent Triangles
21
Example ………
If these polygons are congruent, how do you name them ?
P
O
U
N
M
E
A
T
S
R
1. Pentagon MONTA  Pentagon PERSU
2. Pentagon ATNOM  Pentagon USREP
3. Etc.
Lesson 4-2: Congruent Triangles
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Included Angles & Sides
Included Angle:
A is the included angle for AB & AC.
B is the included angle for BA & BC.
A
*
C is the included angle for CA & CB.
B
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: SSS, SAS, ASA
23
C
Lesson 4-3
Proving Triangles
Congruent
(SSS, SAS, ASA)
Lesson 4-3: SSS, SAS, ASA
24
Postulates
ASA 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.
A
B
SAS
A
D
C
E
F
B
D
C
F
E
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.
Lesson 4-3: SSS, SAS, ASA
25
Postulates
SSS
If the sides of one triangle are congruent to the sides of a
second triangle, then the triangles are congruent.
A
B
D
C E
F
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.
Lesson 4-3: SSS, SAS, ASA
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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?
Lesson 4-3: SSS, SAS, ASA
27
Problem 1 
Given: AB  CD
BC  DA
Prove: ABC  CDA
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA )
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
A
B
1. AB  CD
2. BC  DA
SSS
Reasons
Given
Given
3. AC  CA Reflexive Property
D
C
4. ABC  CDA
Lesson 4-3: SSS, SAS, ASA
SSS Postulate
28
Given : AB  CB ; EB  DB
Problem 2 
Pr ove:
ABE  CBD
Step 1: Mark the Given
Step 2: Mark vertical angles
Step 3: Choose a Method (SSS /SAS/ASA)
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
A
C
B
E
1. AB  CB
2. ABE  CBD
3. EB  DB
D
4. ABE  CBD
Lesson 4-3: SSS, SAS, ASA
SAS
Reasons
Given
Vertical Angles.
Given
SAS Postulate
29
Given : XWY  ZWY ; XYW  ZYW
Problem 3
Pr ove: WXY  WZY
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA)
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Step 6: Is there more?
1. XWY  ZWY
X
W
Y
Z
2. WY  WY
3. XYW  ZYW
4. WXY  WZY
Lesson 4-3: SSS, SAS, ASA
ASA
Reasons
Given
Reflexive Postulate
Given
ASA Postulate
30
Postulates
AAS 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.
A
B
HL
C
E
D
A
D
F
B
C
F
E
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.
Lesson 4-4: AAS & HL Postulate
31
Problem 1 
Given: A  C
BE  BD
Prove: ABE  CBD
Step 1: Mark the Given
Step 2: Mark vertical angles
Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Reasons
Step 6: Is there more?
Given
AAS
A
C
B
E
D
1. A  C
2. ABE  CBD Vertical Angle Thm
3. BE  BD
Given
4. ABE  CBD AAS Postulate
Lesson 4-4: AAS & HL Postulate
32
Given:
Problem 2 
ABC, ADC right
AB  AD
Prove: ABC  ADC
s
Step 1: Mark the Given
Step 2: Mark reflexive sides
Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )
Step 4: List the Parts in the order of the method
Step 5: Fill in the reasons
Statements
Reasons
Step 6: Is there more?
1.
ABC , ADC
Given
HL
A
B
C
right
s
2. AB  AD
D
3. AC  AC
Given
Reflexive Property
4. ABC  ADC HL Postulate
Lesson 4-4: AAS & HL Postulate
33