Download Angles of a Triangle

Triangles and Congruence

§ 5.1 Classifying Triangles

§ 5.2 Angles of a Triangle

§ 5.3 Geometry in Motion

§ 5.4 Congruent Triangles

§ 5.5 SSS and SAS

§ 5.6 ASA and AAS
Classifying Triangles
You will learn to identify the parts of triangles and to
classify triangles by their parts.
In geometry, a triangle is a figure formed when _____
three noncollinear points
are connected by segments.
D
E
F
Each pair of segments forms an angle of the triangle.
The vertex of each angle is a vertex of the triangle.
Classifying Triangles
Triangles are named by the letters at their vertices.
ΔDEF is shown below.
Triangle DEF, written ______,
vertex
angle
D
The sides are: EF, FD, and DE.
The vertices are: D, E, and F.
E
side
F
The angles are: E, F, and D.
In Chapter 3, you classified angles as acute, obtuse, or right.
Triangles can also be classified by their angles.
acute angles.
All triangles have at least two _____
right
obtuse or _____.
The third angle is either _____,
acute ______,
Classifying Triangles
acute
triangle
60°
Triangles
Classified by
Angles
obtuse
right
triangle
120°
triangle
17°
30°
80°
60°
43°
40°
3rd angle is
3rd angle is
3rd angle is
acute
_____
obtuse
______
right
____
Classifying Triangles
scalene
isosceles
equilateral
no
___
sides
congruent
at least two
__________
sides
congruent
all
___
sides
congruent
Triangles
Classified by
Sides
Classifying Triangles
The angle formed by
the congruent sides is called the
vertex angle
___________.
The two angles formed by
the base and one of the
congruent sides are called
base angles
___________.
The congruent sides
are called legs.
leg
leg
The side opposite the vertex
angle is called the _____.
base
Angles of a Triangle
You will learn to use the Angle Sum Theorem.
1) On a piece of paper, draw a triangle.
2) Place a dot close to the center (interior) of the triangle.
3) After marking all of the angles, tear the triangle into three pieces.
then rotate them, laying the marked angles next to each other.
4) Make a conjecture about the sum of the angle measures of the triangle.
Angles of a Triangle
The sum of the measures of the angles of a triangle is 180.
x°
Theorem 5-1
Angle Sum
Theorem
x + y + z = 180
y°
z°
Angles of a Triangle
The acute angles of a right triangle are complementary.
Theorem 5-2
x°
x + y = 90
y°
Angles of a Triangle
The measure of each angle of an equiangular triangle is 60.
x°
3x = 180
Theorem 5-3
x = 60
x°
x°
Geometry in Motion
You will learn to identify translations, reflections, and
rotations and their corresponding parts.
We live in a world of motion.
Geometry helps us define and describe that motion.
In geometry, there are three fundamental types of motion:
reflection and ________.
translation _________,
rotation
__________,
Geometry in Motion
In a translation, you slide a figure from one position to another without
turning it.
Translations are sometimes called ______.
slides
Geometry in Motion
line of
reflection
In a reflection, you flip a figure over a line.
The new figure is a mirror image.
flips
Reflections are sometimes called ____.
Geometry in Motion
In a rotation, you rotate a figure around a fixed point.
turns
Rotations are sometimes called _____.
30°
Geometry in Motion
A
Each point on
the original
figure is called
preimage
a _________.
D
B
C
E
Its matching
point on the
corresponding
figure is called
image
its ______.
F
Each point on the preimage can be paired with exactly one point on its image,
and each point on the image can be paired with exactly one point on its
preimage.
mapping
This one-to-one correspondence is an example of a _______.
Geometry in Motion
A
Each point on
the original
figure is called
preimage
a _________.
B
C
The symbol
→
D
E
Its matching
point on the
corresponding
figure is called
image
its ______.
F
is used to indicate a mapping.
In the figure, ΔABC
→ ΔDEF.
(ΔABC maps to ΔDEF).
In naming the triangles, the order of the vertices indicates the
corresponding points.
Geometry in Motion
A
Each point on
the original
figure is called
preimage
a _________.
D
B
C
E
F
Image
Preimage
D
AB
B
→
→
E
C
→
F
Preimage
A
Its matching
point on the
corresponding
figure is called
image
its ______.
Image
DE
BC
→
→
CA
→
FD
This mapping is called a _____________.
transformation
EF
Geometry in Motion
isometries
Translations, reflections, and rotations are all __________.
An isometry is a movement that does not change the size or shape of the
figure being moved.
When a figure is
translated, reflected, or rotated,
the lengths of the sides of the figure
DO NOT CHANGE.
Congruent Triangles
You will learn to identify corresponding parts of congruent
triangles
If a triangle can be translated, rotated, or reflected onto another triangle, so
congruent triangles
that all of the vertices correspond, the triangles are _________________.
The parts of congruent triangles that “match” are called
corresponding parts
__________________.
vertices indicates the corresponding parts!
The order of the ________
ΔABC  ΔXYZ
Congruent Triangles
In the figure, ΔABC  ΔFDE.
A
As in a mapping, the order of the vertices
_______
indicates the corresponding parts.
C
Congruent Angles
B
Congruent Sides
A  F
AB  FD
B  D
BC  DE
C  E
AC  FE
F
E
These relationships help define the congruent triangles.
D
Congruent Triangles
Definition of
Congruent
Triangles
corresponding parts of two triangles are congruent, then
If the _________________
the two triangles are congruent.
congruent then the corresponding parts
If two triangles are _________,
of the two triangles are congruent.
Congruent Triangles
ΔRST  ΔXYZ.
Find the value of n.
S
Z
50°
R
40°
90°
T
ΔRST  ΔXYZ
S  Y
Y
identify the corresponding parts
corresponding parts are congruent
50 = 2n + 10
subtract 10 from both sides
40 = 2n
divide both sides by 2
20 = n
(2n + 10)°
X
SSS and SAS
You will learn to use the SSS and SAS tests for congruency.
SSS and SAS
4)
CB.
5)
theacute
intersection
F.triangle on
3) Label
AB.
1)
Draw an
scalene
a piece
of paper.
Label its
6)
DF
and
EF. congruent
2)
Construct
a segment
to AC.
Label
the endpoints
of vertices
the
A,
B, and C,
on the
segment
D and
E. interior of each angle.
B
A
C
F
D
E
This activity suggests the following postulate.
SSS and SAS
three
sides of one triangle are congruent to _____
If three _____
corresponding sides of another triangle, then the two
_____________
Triangles are congruent.
S
B
Postulate 5-1
SSS
Postulate
A
C
If AC  RT and AB  RS and
then ΔABC  ΔRST
T
R
BC  ST
SSS and SAS
In two triangles, ZY  FE,
XY  DE, and XZ  DF.
Write a congruence statement for the two triangles.
D
X
Z
Y
Sample Answer:
ΔZXY  ΔFDE
F
E
SSS and SAS
In a triangle, the angle formed by two given sides is called the
included angle of the sides.
____________
C is the included
angle of CA and CB
C
A
A is the included
angle of AB and AC
B
B is the included
angle of BA and BC
Using the SSS Postulate, you can show that two triangles are congruent if their
corresponding sides are congruent. You can also show their congruence
included angle
by using two sides and the ____________.
SSS and SAS
included angle of one triangle are
sides and the ____________
If two
________
congruent to the corresponding sides and included angle of
another triangle, then the triangles are congruent.
S
B
Postulate 5-2
SAS
Postulate
A
C
If AC  RT and A  R and
then ΔABC  ΔRST
T
R
AB  RS
SSS and SAS
On a piece of paper, write your response to the following:
Determine whether the triangles are congruent by SAS.
 If so, write a statement of congruence and tell why they are congruent.
 If not, explain your reasoning.
Q
D
R
P
F
NO! D is not the included angle for DF and EF.
E
ASA and AAS
You will learn to use the ASA and AAS tests for congruency.
ASA and AAS
The side of a triangle that falls between two given angles is called the
included side of the angles. It is the one side common to both angles.
___________
C
AC is the included
side of A and C
CB is the included
side of C and B
A
B
AB is the included
side of A and B
You can show that two triangles are congruent by using _________
two angles and the
included side of the triangles.
___________
ASA and AAS
included side of one triangle are
angles and the ___________
If two
_________
congruent to the corresponding angles and included side of
another triangle, then the triangles are congruent.
S
B
Postulate 5-3
ASA
Postulate
C
A
If A  R and
AC  RT and
then ΔABC  ΔRST
T
R
C  T
ASA and AAS
CA and CB are the nonincluded
sides of A and B
C
A
B
You can show that two triangles are congruent by using _________
two angles and a
nonincluded side
______________.
ASA and AAS
nonincluded side of one triangle are
angles and a ______________
If two
_________
congruent to the corresponding two angles and nonincluded
side of another triangle, then the triangles are congruent.
S
B
Theorem 5-4
AAS
Theorem
C
A
If A  R and
C  T
then ΔABC  ΔRST
R
and CB  TS
T
ASA and AAS
ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to
show congruence.
What other pair of angles must be marked so that the two triangles are
congruent by AAS?
If F and M are marked congruent, then FE and MN would be included
sides.
However, AAS requires the nonincluded sides.
Therefore, D and L must be marked.
D
L
M
F
E
N