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Section 4.1
Congruent Polygons
Polygons
Examples of Polygons
Polygons
Examples of Non-Polygons
Non-Polygons
Naming a Polygon
When naming
polygons, the rule is to
go around the figure,
either clockwise or
counterclockwise, and
list the vertices in
order. It does not
matter which vertex
you list first.
Octagon CDEFGHAB
H
A
G
B
F
C
E
D
Corresponding Sides and Angles
• If two polygons have the same number of
sides, it is possible to set up a correspondence
between them by pairing their parts. In
rhombi RSTU and MNOP, the corresponding
angles and sides would be as follows.
Pair – Angles R and M, S and N, T and O, and U and P
R
U
S
T
M
P
N
O
Polygon Congruence Postulate
Two Polygons are congruent if and only if there
is a correspondence between their sides and
angles such that:
Each pair of corresponding angles is congruent.
Each pair of corresponding sides is congruent.
Otherwise, same shape, same size.
Congruence Statement
• ABCDE ≅ KRMNG, Angle A corresponds to K, B
to R, C to M, D to N, and E to G.
A
N
E
B
D
C
M
G
R
K
Section 4.2
Triangle Congruence
SSS (Side-Side-Side) Postulate
• If the sides of one triangle are congruent to
the sides of another triangle, then the two
triangles are congruent. △VEG ≅△TFH
G
H
12m
12m
19m
V
19m
20m
E
T
20m
F
SAS (Side-Angle-Side) Postulate
• If two sides and their angle in one triangle are
congruent to two sides and their angle in
another triangle, then the two triangles are
congruent. △ABC ≅△FDE
A
F
37˚
37˚
C
B
E
D
ASA (Angle-Side-Angle) Postulate
• If two angles and the side between the two
angles in one triangle are congruent to two
angles and the side between the two angles in
another triangle, then the two triangles are
congruent. △MNO ≅△TSR
M
S
R
39°
56°
O 39°
56°
N
T
Section 4.3
Analyzing Triangle Congruence
AAS (Angle-Angle-Side) Postulate
• If two angles and a non-included side of one
triangle are congruent to two angles and a
non-included side of another triangle, then
the two triangles are congruent. △ABC ≅△EFD
A
101°
C
F
35°
35°
B
D
101°
E
Combinations That Do Not Work
Unless Under Special Circumstances
• AAA (Angle-Angle-Angle)
• SSA (Side-Side-Angle)
When Dealing With Right Triangles
• 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.
• Other Possibilities
• LL (Leg-Leg) Congruence
Theorem
• LA (Leg-Angle)
Congruence Theorem
• HA (Hypotenuse-Angle)
Congruence Theorem
Section 4.4
Using Triangle Congruence
CPCTC in Flowchart Proofs
• CPCTC – Corresponding Parts of Congruent
Triangles are Congruent.
• CPCTC is used after a triangle congruence
postulate (SSS, ASA, SAS, or AAS) has been
establish to prove two triangles are congruent.
• Flowchart Proofs are used to explain and
understand why two or more triangles are
congruent.
Given: AC≅BD, CX≅DX, and <C≅<D
Prove: X is the midpoint of AB.
C
B
X
A
D
AC≅BD
Given
X is the
midpoint
of AB by
Def. of
midpoint
AX≅BX by
CPCTC
△ACX≅△BDX
by SAS
CX≅DX
Given
<C≅<D
Given
Isosceles Triangle
• An isosceles triangle is a triangle with two
congruent sides.
• The two congruent sides are known as the
legs of the triangle.
• The remaining side is known as the base.
• The angle opposite the base is the vertex
angle.
• The angles whose vertices are the endpoints
of the base are base angles.
Theorems Involving Isosceles Triangles
• Isosceles Triangle Theorem – If two sides of a
triangle are congruent, then the angles
opposite those sides are congruent.
• Converse of the Isosceles Triangle Theorem –
If two angles of a triangle are congruent, then
the sides opposite those angles are congruent.
• The bisector of the vertex angle of an isosceles
triangle is the perpendicular bisector of the
base.
Reminders For an Equilateral Triangle
• The measure of each angle of an equilateral
triangle is 60°.
• The measures of the sides are congruent.
Section 4.5
Proving Quadrilateral Properties
Quadrilateral Hierarchy
Quadrilateral –
polygon with 4
sides
Quad’s with no
pairs of parallel
sides
Kites
Trapezoids - one
pair of parallel
sides
Isosceles
Trapezoids
Parallelograms two pair of
parallel sides
Rectangles
Squares
Rhombi
Properties of Parallelograms
• A diagonal divides the
parallelogram into two
congruent triangles.
• Opposite sides of a
parallelogram are ≅.
• Opposite angles of a
parallelogram are ≅.
• Consecutive angles are
supplementary.
• The two diagonals
bisect each other.
• Rectangles, rhombi, and
squares are
parallelograms.
Properties of Parallelograms
Properties of Rectangles
• The diagonals are ≅.
• Four 90° angles.
Properties of Rhombi
• The diagonals are
perpendicular.
• Four ≅ sides.
• The diagonals bisect a pair
of opposite angles.
Properties of Squares
• All properties of a parallelogram
• All properties of a rectangle
• All properties of a rhombus
Section 4.6
Conditions for Special Quadirlaterals
The Conditions That Determine a
Figure
• If you are given a quadrilateral, check the
given information against the definitions of
the special figures.
• Check for properties that match or you can
prove to a special figure.
• It is better to be more specific than general.
Remember a square is always a parallelogram
but a parallelogram is not always a square.
The Triangle Midsegment Theorem
• A midsegment of a triangle is a parallel to a
side of the triangle and has a measure equal
to half of the measure of that side.
• CD is the midsegment of △AGH.
G
H
C
D
A
Review of Other Properties
•
•
•
•
•
•
•
•
Substitution Property
Transitive Property
Reflexive Property
Symmetric Property
Transitive Property
SSS (Side-Side-Side)
ASA (Angle-Side-Angle)
SAS (Side-Angle-Side)
•
•
•
•
•
•
•
•
AAS (Angle-Angle-Side)
CPCTC
Definition of Midpoint
Alt. Interior Angles
Cons. Interior Angles
Corresponding Angles
Alt. Exterior Angles
Vertical Angles Theorem
Section 4.7
Compass and Straightedge
Constructions
Compass and Straightedge
Constructions
• A segment congruent to • The perpendicular
a given segment pg. 261
bisector of a given
segment and the
• A triangle congruent to
midpoint of a given
a given triangle pg. 262
segment pg. 266
• Angle bisector pg. 263
• An angle congruent to a • A line through a point
perpendicular to a given
given angle pg. 265
line pg. 267
• A line through a point
parallel to a given line
pg. 268
Section 4.8
Constructing Transformations
Transformations
• Rotation (Turn) – every point of the preimage
is rotated by a given angle about a point.
• Reflection (Flip) – every point of the preimage
may be connected to its image point by a
segment that (a) is perpendicular to the line
or plane that is the mirror of the reflection
and (b) has its midpoint on the mirror of the
reflection.
Transformations
• Translation (Slide) – every point of the
preimage moves in the same direction by the
same amount to form the image.
• Dilation – a figures size is increased or
decreased by a scale factor, but its shape stays
the same.
Other Theorems
Betweenness Postulate
• Converse of the Segment
Addition Postulate
• Given three points P, Q, and
R, if PQ + QR = PR, then P,
Q, and R are collinear and Q
is between P and R.
Triangle Inequality Theorem
• The sum of the lengths of
any two sides of a triangle is
greater than the length of
the third side.
• AB + BC > AC
A
• AB + AC > BC
• AC + BC > AB
B
C