Download Chapter 4 - Catawba County Schools

Congruent Triangles
Geometry Chapter 4
4.1 Triangles and Angles

Classification by Sides:
Triangles and Angles

Classification by Angles
Parts of Triangles
Interior
angle
Exterior
angle
Vertex angle
leg
leg
hypotenuse
leg
Base angle
Base angle
base
leg
Theorems Involving Triangles


The sum of the
measures of the
angles of a triangle =
180°
The measure of the
exterior angle of a
triangle = the sum of
the two remote
interior angles.
B
C
A
3
2
1
Corollaries to Triangle
Theorems



The acute angles of a right
triangle are complementary.
Each angle of an equiangular
triangle has a measure of
60°.
In a triangle, there can be at
most one right angle or one
obtuse angle.
¬
Examples

Sides of lengths 2mm, 3mm and 5mm.

Sides of lengths 3m, 3m, 3m.

Sides of lengths 8m, 8m, 5m.
Examples

Angles of measures 90, 25, 65.

Angles of measures 60, 60, 60.

Angles of measures 80, 70, 30.

Angles of measures 140, 30, 10.
Examples

A triangle has angles that measure x, 7x,
and x. Find x.
Examples

A right triangle has angle measures of x
and (2x-21). Find x.
Examples

Find the measure of the exterior angle
shown.
4.2 Congruence and Triangles


Congruent – same
size, same shape
Congruent
Polygons(Triangles)
– Two polygons
(triangles) are
congruent iff their
corresponding sides
and corresponding
angles are congruent
E
B
A
If ΔABC 
then
A  D
B  E
C  F
C
D
ΔDEF,
AB  DE
BC  EF
AC  DF
F
Theorems about Congruent
Figures

If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are
congruent.
S
N
R

If  R   M and  S
then  T   O
T

N,
M
O
Examples
L
110°
H
M
G
(2x +3)m
(7y + 9)°
87°
O
72°
10m
N
E
If LMNO  EFGH, find
x and y.
F
Examples
4.3-4.3 Proving Triangles
Congruent

SSS – Side Side
Side – If three sides
of one triangle are
congruent to three
sides of another
triangle, then the
triangles are
congruent.
If AB  DE
BC  EF
AC  DF, then
ABC  DEF
SAS

SAS – Side Angle Side
– 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.
If AB  DE
BC  EF
B  E,
then
ABC  DEF
ASA

ASA – Angle Side Angle –
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 congruent.
If A  D
C  F
AC  DF,
then
ABC  DEF
AAS

AAS – Angle Angle Side
– If two angles and a
non-included side of one
triangle are congruent to
two angles and the
corresponding nonincluded side of another
triangle, then the
triangles are congruent.
If A  D
C  F
AB  DE,
then
ABC  DEF
HL

HL – Hypotenuse Leg
– If the hypotenuse
and leg of one RIGHT
triangle are congruent
to the hypotenuse and
leg of another RIGHT
triangle then the
triangles are
congruent.
D
A
C
B
F
If ABC,DEF Right s,
AB  DE, AC  DF,
then
ABC  DEF.
E
4.5 Using Congruent Triangles

Definition of Congruent Triangles
(rewritten)
Corresponding Parts of
Congruent
Triangles are Congruent
CPCTC is used often in proofs involving
congruent triangles.
M
A is the midpoint of MT.
A is the midpoint of SR.
MS ll TR
1. A is the midpoint of MT.
A is the midpoint of SR.
R
A
S
1. Given
T
UR ll ST
R and T are right angles
U
R
1. UR ll ST
1. Given
R and T are right angles
T
S
4.6 Isosceles, Equilateral and
Right Triangles


B
If two sides of a triangle
are congruent, then the
angles opposite are
congruent. (Base angles
of an isosceles triangle
are congruent.
Converse – If two angles
of a triangle are
C
A
congruent, then the sides If BA  BC, then A  C.
opposite are congruent.
If A  C, then BA  BC.
More Corollaries

B

A
C
If a triangle is
equilateral, then it is
equiangular.
If a triangle is
equiangular then it is
equilateral.
Examples

Find x and y.
y
35
x
Examples

Find the unknown measures.
?
50
?
Examples

Find x.
(x-11) in
33 in
Examples

Find x and y.
y
40
x