Download Chapter 4 Notes

Chapter 4 Notes
4.1 – Triangles and Angles
B
A
C
Scalene – No
congruent sides
A Triangle  Three segments
joining three noncollinear
points. Each point is a
VERTEX of the triangle.
Segments are SIDES!
Isosceles – At
LEAST 2
congruent sides
Equilateral – All
sides congruent
Acute – 3 acute
angles
Right – One right
angle
Obtuse – One
obtuse angle
Equiangular – all
angles congruent
B
C
A
A is opposite the side BC
Side AB is opposite C
opposite means non - adjacent
TRIANGLE SUM THEOREM
The sum of the measures of the angles
of a triangle is 180.
2
1
3
Given : ABC
Prove : m1  m2  m3  180
Exterior Angles Theorem
The measure of an exterior angle of a triangle
equals the sum of the two remote interior
angles. (remote means nonadjacent)
Statement
3
1 2
4
Given : A triangle with exterior
angle 1
Prove : m1  m3  m4
Reason
Corollary to triangle sum theorem: Acute angles of a
right triangle are complementary.
All angles 180, if one is 90, the other two add up to
90, and are complementary
4.2 – Congruence and
Triangles
When TWO POLYGONS have the same size
and shape, they are called CONGRUENT!
Their vertices and sides must all match up to
be congruent.
When two figures are congruent, their
corresponding sides and corresponding
angles are congruent. Identical twins!
Name all the corresponding
parts and sides, then make
a congruence statement.
B
AB  FE A  F
BC  ED B  E
CA  DF C  D
ABC  FED
BCA  EDF
CAB  DFE
A
D
E
C
F
If you notice, the way you name the triangle is
important, all the CORRESPONDING SIDES must
line up!
3rd Angles Theorem: If two angles of one triangle
are congruent to two angles of another triangle, then
the 3rd angles are congruent.
B
If A  D, B  E
Then C  F
A
E
C
D
F
E
A
G
80
5 y  10
M 45
15
H
3x
Find x
T
Y
24
Find y
O
• Note, triangles also have the following
properties of congruent: Reflexive, symmetric,
and transitive.
Reflexive : ABC  ABC
Symmetric : ABC  DEF , DEF  ABC
Transitive : ABC  DEF, DEF  XYZ then
ABC  XYZ
4.3 – 4.4 Proving Triangles
are Congruent
A
D
If AB  EF , BC  DE , AC  DF
Then ABC  FED
B
E
C
F
SSS Congruence Postulate –
If three sides of one triangle
are congruent to three sides
of a second triangle, then the
two triangles are congruent.
SAS Congruence Postulate –
If two sides and the included
angle of one triangle are
congruent to two sides and
the included angle of a
second triangle, then the two
triangles are congruent.
A
D
B
E
If BC  DE , C  D, AC  DF
Then ABC  FED
Included means IN
BETWEEN
C
F
A
D
If A  F , AC  DF , C  D
Then ABC  FED
B
E
C
F
ASA Congruence Postulate –
If two angles and the
included side of one triangles
are congruent to two angles
and the included side of a
second triangle, then the two
triangles are congruent.
AAS Congruence Theorem – If
two angles and a nonincluded
side of one triangle are
congruent to two angles and
the corresponding
nonincluded side of a second
triangle, then the two trianges
are congruent.
If B  E, C  D, AC  DF
Then ABC  FED
A
D
B
E
C
F
A
D
B
E
C
F
A
B
C
E
F
Helpful things for the future!
Reflexive sides
Reflexive angles
E
B
G
A
D
BC  BC
H
F
C
E  E
When you see shapes sharing a side, you
state that fact using the reflexive property of
congruence!
A
B
D
C
Draw and write down if the triangles are
congruent, and by what thrm\post
Proofs! The way I like to think about it to look at all the
angles and sides, and don’t be fooled by the picture.
A
D
1. C is the midpoint 1. Given
C
of AE and BD
B
E
Given : C is the midpoint
of AE and BD
Prove : ABC  EDC
Tips, label the diagram as you go
along.
A
B
D
Given : AD  BE , AB  DE
Prove : ABD  EDB
E
1. AD  BE
AB  DE
1. Given
Given : EF  EH , EB  EC
Prove : EBH  ECF
E
B
G
What about the angle?
F
1) EF  EH , EB  EC
C
1) Given
H
Use SSS Congruence Postulate to
show that ABC  DEF
(-5, 1)
A
B
(-4, -3)
AB 
BC 
DE 
AC 
DF 
EF 
(1, 3)
E
(5, 4)
D
F(2, 2)
C
(-3, -2)
A
B
D
E
1. AD || BE
AB || DE
Given : AD || BE , AB || DE
Prove : ABD  EDB
Tips, label the diagram as you go
along.
1. Given
Given : DU || KC
DU  KC; D  K
Prove : DUC  KCS
K
D
U
1. DU || KC; DU  KC; D  K 1. Given
C
S
Given : AC || BE , BC || DE , AC  BE
Prove : ABC  BDE
A
B
D
C
E
Given : B  E, BC  CE
Prove : ABC  DEC
B
A
C
E
D
4.5 – Using Congruent
Triangles
CPCTC  Correspond ing Part of Congruent Triangles
are Congruent  Corresp Parts of  ' s are 
A
D
1. C is the midpoint 1. Given
C
B
E
Given : C is the midpoint
of AE and BD
Prove : AB || ED
of AE and BD
2. AC  EC
2. Def of mdpt
BC  DC
3. ACB  ECD 3. VAT
4. ACB  ECD 4. SAS  post
Some Ideas that may help you.
If they want you to prove something, and you see
triangles in the picture, proving triangles to be congruent
may be helpful.
If they want parallel lines, look to use parallel line
theorems (CAP, AIAT, AEAT, CIAT)
Know definitions (Definition of midpoint, definition of
angle bisectors, etc.)
Sometimes you prove one pair of triangles are
congruent, and then use that info to prove another
pair of triangles are congruent.
A
D
Given : AD  BE , AB  DE
Prove : BE || AD
B
E
1. AD  BE
AB  DE
1. Given
Given : DU || KC
K
D
DU  KC; D  K
Prove : C is the mdpt of US
U
1. DU || KC; DU  KC; D  K 1. Given
C
S
Given : B  E, BC  CE
Prove : AC  DC
B
A
C
E
D
Given : ALG  ALN
 LGA  LNA
N
L
A
E
Prove : ELN  ELG
G
ALG  ALN,  LGA  LNA
Given
You try this classic proof!
Given : 1  2, DE  BE
Prove : AC is the angle
bisector of DCB
D
A 3
4
1
2 E
B
5
6
C
4.6 – Isosceles, Equilateral,
and Right Triangles.
• Bring book Tuesday
• We will go over what’s going to be on
Wednesday’s Quiz at end of Tuesday
lesson
Vertex Angle
LEGS
BASE
Base Angles
Remember,
definition of
isosceles
triangles is
that AT LEAST
two congruent
sides.
Base angles theorem – If two sides of a
triangle are congruent, then the base angles
are congruent.
Converse of Base angles
theorem – If base angles are
congruent, then the two
opposite sides are congruent.
Corollary 1 – An equilateral triangle is also
equiangular (Use isosceles triangle theorem multiple
times with transitive)
Corollary 2 – An equilateral triangle has three 60
degree angles (Use corollary 1 and angle of triangle
equals 180)
Hypotenuse Leg Theorem
(HL) – If the hypotenuse
and ONE of the legs of a
RIGHT triangle are
congruent, then the
triangles are congruent.
A
B
D
C
Draw and write down if the triangles are
congruent, and by what thrm\post
D
Given : DUK is an isosceles triangle with
DU and DK as legs.
1 and 2 are right angles.
Pr ove : U  K
1 2
U
C
K
DU  DK
Def of isosceles triangle
Given : AB  AD, AC  BD
Prove : ABC  ADC
A
B
C
D
4.7 – Triangles and
Coordinate Proof
Given a right triangle with one vertex (-20, -10), and legs
of 30 and 40, find two other vertices, then find the length
of the hypotenuse.
Given a vertex of a rectangle at the origin, find three
other possible vertices if the base is 15 and the height is
10 for a rectangle. Then find the area.
Given the coordinates, prove that the AC is the angle
bisector of BCD
B
A
C
D
(d, k)
(a, b)
(__,__)
(__,__)
(__,__)
(__, k)
(h,k)
(j,__)
(a,__)
(__,__)
Picking convenient variable coordinates, prove
that the diagonals of a rectangle are congruent.