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4.1 Triangles and Angles
Triangle: a figure formed by three segments joining three noncollinear
points. It can be classified by its sides and by its angles.
Classification by Sides
Equilateral
Isosceles
3 congruent sides
at least 2 congruent sides no congruent sides
Classification by Angles
Acute
Equiangular
3 acute angles
Scalene
3 congruent angles
Right
Obtuse
1 right angle
1 obtuse angle
Vertex: each of three points joining the sides of a triangle.
A, B, and C are vertices of triangle ABC.
A
B
C
Adjacent sides: two sides sharing a common vertex. AB and BC share vertex B.
Right and Isosceles Triangles
Right Triangle
Isosceles Triangle
hypotenuse
leg
leg
leg
leg
base
Triangle Sum Theorem:
The sum of the measures of the interior angles of a triangle is 180*.
m<A + m<B + m<C = 180º
A
B
C
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two nonadjacent interior angles.
m<1 = m<A + m<B
A
B
C
Corollary to the Triangle Sum Theorem:
The acute angles of a right triangle are complementary.
m<A + m<B = 90º
A
C
Classwork: page 198: 1-9
Homework: page 198: 10 – 26, 29 – 39
B
4.2 Congruence and Triangles
Congruent: two geometric figures that have exactly the same size and
shape. When two figures are congruent, there is a correspondence
between their angles and sides such that corresponding sides are
congruent and corresponding angles are congruent.
ABC  PQR
A
B
Corresponding Angles
<A  < P
<B  < Q
< C < R
P
C
R
Corresponding Sides
AB  PQ
BC  QR
CA  RP
Q
Third Angle Theorem:
If two angles of one triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
If <A  <P and <B  <Q, then <C  <R
Reflexive Property of Congruent Triangles:
Every triangle is congruent to itself.
Symmetric Property of Congruent Triangles:
If ABC DEF, then DEF  ABC.
Transitive Property of Congruent Triangles:
If ABC  DEF and DEF  JKL, then ABC  JKL
Classwork: page 205: 1 – 9.
Homework: 206: 10 – 22, 24 – 29
4.3 Proving Triangles are Congruent: SSS and SAS
SSS Congruence Postulate
If three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
If Side AB  DE
C
F
Side BC  EF
Side CA  FD
Then ABC  DEF
A
B
E
D
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.
If Side AB  DE
C
F
Angle <B  <E
Side BC  EF
Then ABC  DEF
A
B
E
D
Choosing Which Congruence Postulate to Use
D
F
E
A
B
C
G
H
L
X
W
N
M
P
Q
Z
Y
Name the included angle between the given pair of sides.
A
AB and BC
C
B
D
CE and DC
AC and BC
E
Classwork: page 216: 2 – 11
Homework: page 216: 12 – 22
SSS Congruence Postulate
If three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
If Side AB  DE
C
F
Side BC  EF
Side CA  FD
Then ABC  DEF
A
B
E
D
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.
If Side AB  DE
C
F
Angle <B  <E
Side BC  EF
Then ABC  DEF
A
B
E
D
4.4 Proving Triangles are Congruent: ASA and AAS
ASA Congruence Postulate
If two angles and the included side on one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles
are congruent.
If Angle <C  <F
C
F
Side BC  EF
Angle <B  <E
Then ABC  DEF
A
B
E
D
AAS Congruence Postulate
If two angles and the nonincluded side of one triangle are congruent to
two angles and the nonincluded side of a second triangle, then the two
triangles are congruent.
If Angle <C  <F
C
F
Angle <A  <D
Side AB  DE
Then ABC  DEF
A
B
E
D
D
F
E
G
A
B
C
H
Q
X
T
R
S
W
Z
Y
4.5 Using Congruent Triangles
Knowing that all pairs of corresponding parts of congruent triangles are
congruent can help one prove congruent parts of triangles.
Given: PS  RS, PQ  RQ
Prove: <PQS  <RQS
Statements
Reasons
1. PS  RS
1. Given
2. PQ  RQ
2. Given
3. QS  QS
3. Reflexive
4. PQS  RQS
4. SSS
5. <PQS  <RQS
5. CPCTC
Q
P
R
S
Once one proves two triangles are congruent, then any pair of congruent
parts are congruent by CPCTC: Corresponding Parts of Congruent
Triangles are Congruent. These may be corresponding sides or angles.
Classwork: page 232: 1 - 3
Assignment: 232: 4 – 18, 22 – 23 all
4.6 Isosceles, Equilateral, and Right Triangles
B
Base Angle Theorem: if two sides of a triangle are
A
congruent, then the angles opposite them are congruent.
If AB  AC, then <B  <C
Base Angle Converse: if two angles of a triangle are
Congruent, then the sides opposite them are congruent.
If <B  <C, then AB  AC
C
B
A
C
Corollary: if a triangle is equilateral, then it is equiangular.
Corollary: if a triangle is equiangular, then it is equilateral.
HL: Hypotenuse-Leg: if the hypotenuse and a leg
of one right triangle are congruent to the hypotenuse
and a leg of a second right triangle, then the two
triangles are congruent.
In the two right triangles
if leg BC  EF
and leg AC  DF
then ABC   DEF by HL
Triangle may be proved congruent by any of five ways
SSS
SAS
ASA
AAS
HL
Page 289: 1 – 25, 29 – 37 all