Download 4.1- 4.4 - Fulton County Schools

Stuck on 4.1 – 4.4?
Katalina Urrea and Maddie Stein ;)
Vocabulary
• Base angle- angles whose vertices are the endpoints of
the base
• Base of an isosceles triangle- the angles whose vertices
are the endpoints of the base of an isosceles triangle
• CPCTC- Abbreviation for “corresponding parts of
congruent triangles are congruent”
• Corollary- A theorem that follows directly from another
theorem and that can easily be proved from that
theorem
• Isosceles triangle- A triangle with at least two
congruent sides
• Legs of an isosceles triangle- The two congruent sides
of an isosceles triangle
• Vertex angle- The opposite angles formed by two
intersecting lines.
4.1
Congruent
Polygons
Polygon Congruence Postulate
•
•
Two polygons are congruent IFF (if and
only if) there is a correspondence
between their sides and angles such
that:
-Each pair of corresponding angles are
congruent
-Each pair of corresponding sides are
congruent
(Converse is true as well)
Naming Polygons
• You must name polygons in order
• The name of this polygon
is ABCDEF
• You can also name it
BCDEFA, CDEFAB
and so on, but you MUST
keep it in order.
A
B
F
C
E
D
Side and Angle Congruence
E
A
ABCD
EFGH
H
D
G
B
F
C
Sides:
Angles:
AB
EF
<A
<E
BC
FG
<B
<F
CD
GH
<C
<G
DA
HE
<D
<H
4.2
Triangle
Congruence
Side-Side-Side Postulate (SSS)
• If the sides of one triangle are congruent
to the sides of another triangle then those
triangles are congruent.
A
Given: ABCD is a rhombus
Prove:
ABD
DBC
B
D
C
Statements
Reasons
ABCD is rhombus
Given
AB
BC
BD
BD
ABD
CD
DA
Definition of Rhombus
Reflexive
DBC
SSS
Side-Angle-Side Postulate (SAS)
• If two sides and the included angle in one
triangle are congruent to two sides and the
included angle in another triangle, then
those two triangles are congruent.
A
B
Given: AB//CD AB
CD
Prove:
D
C
Statements
AB//CD AB
<BDC
DB
ABD
ABD
CBD
Reasons
CD
<ABD
DB
Given
Alternate Interior Angle
Reflexive
CBD
SAS
Angle-Side-Angle Postulate (ASA)
• If two angles and the included side of a
triangle are congruent to two angles and
an included side of another triangle, then
the two triangles are congruent.
D
Given: <A
Prove:
<E
AC
ABC
E
C
B
A
Statements
<A
<E AC
CE
<ACB
<DCB
ABC
CDE
Reasons
Given
Vertical Angles
ASA
CE
CDE
4.3
Angle-Angle-Side Theorem (AAS)
• If two angles and a non-included side of
one triangle are congruent to the
corresponding angles and non-included
side of another triangle, then the triangles
are congruent.
B
C
Given: AD
Prove:
BAD
AE
<C
CAE
F
E
D
A
Statements
AD
AE <C
<DAB
<EAC
BAD
CAE
<B
Reasons
Given
Reflexive
AAS
<B
HL (Hypotenuse-Leg) Congruence
Theorem
• If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and a leg of another right triangle, then the
two triangles are congruent.
B
Given:
ABC is isosceles
BD perpendicular CA
Prove:
A
ABD
CBD
C
D
Statements
ABC is isosceles
BD perpendicular CA
AB BC
<BDA= 90°
<BDC=90°
<BDA
<BDC
BD
BD
ABD
CBD
Reasons
Given
Given
Definition of Isosceles
Definition of Perpendicular
Definition of Perpendicular
Transitive
Reflexive
4.4
Isosceles
Triangles
Isosceles Triangle Theorem (Base
Angle Theorem)
• If two sides of the triangle are congruent,
then the two angles opposite those sides
are congruent.
• The converse is also true.
B
A
Given: AB
BC
Prove: <A
<B
C
D
Statements
AB
BC
DB is an angle bisector
<ABD
<CBD
DB
DB
ABD
CBD
<A
<B
Reasons
Given
Construction
Definition of Angle Bisector
Reflexive
SAS
CPCTC
Corollaries
1) The bisector of the vertex angle of an
isosceles triangle is the perpendicular
bisector of the base.
2) The measure of each angle in an
equilateral triangle is 60°.
• http://www.washoe.k12.nv.us/ecollab/was
hoemath/dictionary/vmd/full/s/side-sidesidesss.htm
• http://www.ekacademy.org/mines/hspe/Cr
eateHtm/htm/4-8-2_n-nevadan-4-2-31.htm