Download Chapter 4: Congruent Triangles

Chapter 4:
Congruent Triangles
4-1 Congruent Figures
Congruent- when two figures have the same size and
shape
D
A
B
C
E
F
4-1 Continued
Congruent triangles- two triangles are congruent if
and only if their vertices can be matched up so
that the corresponding parts (angles and sides) of
the triangle are congruent
1.
2.
Their corresponding angles are congruent
because congruent triangles have the same
shape.
Their corresponding sides are congruent because
congruent triangles have the same size.
4-1 Continued


Congruent parts of triangles are marked alike.
Congruent triangles must be named in the same
order of congruency.
S
N
SUN
U
A
Y
R
RAY
4-1 Continued
When justifying statements by use of the definition of
congruent triangles, use this wording:
Corresponding parts of congruent triangles are
congruent, which is written:
Corr. Parts of
s are .
4-1 Continued
Congruent polygons- two polygons are congruent if
and only if their vertices can be matched up so that
their corresponding parts are congruent
C
B
A
H
D
F
G
ABFGH
E
BCDEF
4-2 Some Ways to Prove
Triangles Congruent
Proving triangles congruent
with only three
corresponding parts.
1. Side Side Side Postulate
(SSS)- if three sides of one
triangle are congruent to
three sides of another
triangle, then the triangles
are congruent
GOB
L
G
O
B E
LET by the SSS Postulate
T
4-2 Continued
Side Angle Side Postulate
(SAS)- if two sides and the
included angle of one
triangle are congruent to
two sides and the included
angle of another triangle,
then the triangles are
congruent
JEN
P
J
E
N A
PAK by the SAS Postulate
K
4-2 Continued
Angle Side Angle Postulate
(ASA)- if two angles and
the included side of one
triangle are congruent to
two angles and the included
side of another triangle,
then the triangles are
A
congruent
CAR
O
C
R L
OLY by the ASA Postulate
Y
Proof of ASA Postulate
Statement
Given: E is the midpoint of
1. E is the midpoint of
2.
Prove:
3.
4.
5.
T
M
E
6.
J
Reason
1. Given
2. Definition of a midpoint
3. Given
4. If two lines are perpendicular
then they form congruent adjacent
angles.
5. Reflexive property of congruence
6. SAS postulate
4-3 Using Congruent Triangles
Learning how to extract information on segments or
angles once it is shown that they are corresponding
parts of congruent triangles…
4-3 Continued
Statement
1.
Given: AB and CD bisect
each other at M
and
bisect each other at M
2. M is the midpoint of
and of
Prove: AD ll BC
;
6.
M
D
7.
B
2. Definition of a bisector of a
segment
4. Vertical angles are congruent
5. SAS Postulate
5.
C
1. Given
3. Definition of a midpoint
3.
4.
A
Reason
ll
6. Corresponding parts of
congruent triangles are
congruent
7. If two lines are cut by a
transversal and alternate
interior angles are congruent,
then the lines are parallel.
4-3 Continued
A line and a plane are perpendicular if and only if they
intersect and the line is perpendicular to all lines in the
plane that pass through the point of intersection.
P
X
O
4-3 Continued
Statement
1.
2.
Given: PO plane X;
3. m
AO BO
Prove: PA
P
X
A
O
B
plane X
1. Given
;
= 90; m
4.
Reason
= 90
2. Definition of a line
perpendicular to a plane.
5.
3. Definition of perpendicular
lines
6.
4. Defintion of congruent angles
7.
5. Given
8.
6. Reflexive Property
7. SAS postulate
8. Corresponding parts of
congruent angles are
congruent.

4-3 Continued
To prove two segments or two angles are
congruent:
1.) Identify two triangles in which the two
segments or angles are corresponding parts.
2.) Prove that the triangles are congruent.
3.) State that the two parts are congruent, using
this reason
Corr. Parts of
s are
.
4-4 The Isosceles Triangle
Theorems
Legs- the congruent sides of a
triangle
Base- the non-congruent side
of a triangle
Base angles- the angles at
the base of the triangle
Vertex angle- the angle
opposite the base of the
isosceles triangle
Vertex angle
Leg
Leg
Base angles
Base
4-4 Continued
The Isosceles Triangle Theorem- if two sides of a
triangle are congruent, then the angles opposite those
sides are congruent
A
D
C
B
4-4 Continued
Corollary 1- an equilateral triangle is also
equiangular
Corollary 2- an equilateral triangle has three 60
degree angles
Corollary 3- The bisector of the vertex angle of an
isosceles triangle is perpendicular to the base at its
midpoint
4-4 Continued
Theorem 4-2
If two angles of a triangle
are congruent, then the
sides opposite those angles
are congruent.
Corollary- an equilateral
triangle is also equilateral
A
D
*
C
Theorem 4-2 is the converse of Theorem
4-1, and the corollary of Theorem 4-2 is
the converse of Corollary 1 of Theorem
4-1.
B
4-5 Other Methods of Proving
Triangles Congruent
Angle Angle Side Theorem (AAS)- if two angles and
a non-included side of one triangle are congruent to
the corresponding parts of another triangle, then the
triangles are congruent
Y
A
E
C
U
O
4-5 Continued
Hypotenuse- the side opposite the right angle in a
right triangle
Legs- the other two sides of the triangle
hypotenuse
leg
leg
4-5 Continued
Hypotenuse Leg Theorem- if the hypotenuse and a
leg of one right triangle are congruent to the
corresponding parts of another right triangle, then the
triangles are congruent
O
C
A
T G
B
N
4-5 Continued
Leg-Leg Method- if two legs of one right triangle are
congruent to the two legs of another right triangle,
then the triangles are congruent
Hypotenuse-Acute Angle Method- if the hypotenuse
and an acute angle of one right triangle are congruent
to the hypotenuse and an acute angle of another right
triangle, then the triangles are congruent
Leg-Acute Angle Method- If a leg and an acute angle
of one right triangle are congruent of the corresponding
parts in another right triangle, then the triangles are
congruent.

4-6 Using More than One Pair of
Congruent Triangles
Statement
Given:
1.
Prove:
;
2.
3.
4.
5.
6.
7.
B
8.
A
1
2
3
O
4
D
5
6
C
Reason
1. Given
2. Reflexive property
3. ASA postulate
4. Corresponding parts of
congruent angles are
congruent.
5. Reflexive property
6. SAS postulate (1, 4, 5)
7. Corresponding parts of
congruent angles are
congruent.
8. If two lines form congruent
adjacent angles, then the
lines are perpendicular.
4-7 Medians, Altitudes, and
Perpendicular Bisectors
Median- a segment from a vertex to the midpoint of
the opposite side in a triangle
B
B
B
A
C
A
A
C
C
4-7 Continued
Altitude- the perpendicular segment from a vertex to a
line that contains the opposite side
In an acute triangle, the three altitudes are all inside the
right triangle.
B
A
B
B
A
C
A
C
C
4-7 Continued
A
A
A
B
B
C
B
C
C
In a right triangle, two of the altitudes are parts of the triangle.
They are the legs of the right triangle. The third altitude is inside
the triangle.
A
A
A
L
B
J
B
C
B
C
K
In an obtuse triangle, two of the altitudes are
outside the triangle.
C
4-7 Continued
Perpendicular bisector- a line (or ray or segment)
that is perpendicular to the segment at its midpoint
M
N
p
O
4-7 Continued
Theorem 4-5
If a point lies on the perpendicular bisector of a segment, then
the point is equidistant from the endpoints of the segment.
I
K
L
t
J
4-7 Continued
Theorem 4-6
If a point is equidistant from the endpoints of a
segment, then the point lies on the perpendicular
bisector of the segment.
*Theorem 4-6 is the converse of Theorem 4-5.
I
1
K
2
L
J
4-7 Continued
The distance from a point to a line (or plane) is
defined to be the length of the perpendicular
segment from the point to the line (or plane).
B
A
D
R
t
4-7 Continued
Theorem 4-7
If a point lies on the
bisector of an angle,
then the point is
equidistant from the
sides of the angle.
A
E
G
F
B
D
C
4-7 Continued
Theorem 4-8
If a point is equidistant
from the sides of an angle,
then the point lies on the
bisector of the angle.
* Theorem 4-8 is the converse
of Theorem 4-7.
K
I
M
N
J
H
L
The End
(Thank God!)