1. 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

2. 5-Minute Check

3. Classifying Triangles
What You'll Learn
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 _____ noncollinear points
are connected by segments.
three E D F
Each pair of segments forms an angle of the triangle.
The vertex of each angle is a vertex of the triangle.
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4. Classifying Triangles
Triangles are named by the letters at their vertices.
Triangle DEF, written ______, is shown below.
ΔDEF
vertex
The sides are:
EF, FD, and DE. E D F
angle
The vertices are:
D, E, and F.
The angles are:
E, F, and D.
side
In Chapter 3, you classified angles as acute, obtuse, or right.
Triangles can also be classified by their angles.
All triangles have at least two _____ angles.
acute
The third angle is either _____, ______, or _____.
acute
obtuse
right

5. Classifying Triangles
Classified by
Angles
acute
triangle
obtuse
triangle
right
triangle
60°
80°
120°
17°
43°
30°
60°
40°
3rd angle is
_____
3rd angle is
______
3rd angle is
____
acute
obtuse
right

6. Classifying Triangles
Classified by
Sides
scalene
isosceles
equilateral
___
sides
congruent no
__________
sides
congruent
at least two
___
sides
congruent
all
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7. What You'll Learn You will learn to use the Angle Sum Theorem.
Angles of a Triangle
What You'll Learn
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.
1 - 7

8. The sum of the measures of the angles of a triangle is 180.
Theorem 5-1
Angle Sum
Theorem
The sum of the measures of the angles of a triangle is 180. z° x° y°
x + y + z = 180

9. The acute angles of a right triangle are complementary.
Angles of a Triangle
Theorem 5-2
The acute angles of a right triangle are complementary. x° y°
x + y = 90

10. The measure of each angle of an equiangular triangle is 60.
Angles of a Triangle
Theorem 5-3
The measure of each angle of an equiangular triangle is 60. x°
3x = 180
x = 60
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11. Geometry in Motion
What You'll Learn
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:
__________, _________, and ________.
translation
reflection
rotation
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12. Geometry in Motion
Translation
In a translation, you slide a figure from one position to another without turning it.
Translations are sometimes called ______.
slides

13. Reflection In a reflection, you flip a figure over a line.
Geometry in Motion
line of
reflection
Reflection
In a reflection, you flip a figure over a line.
The new figure is a mirror image.
Reflections are sometimes called ____.
flips

14. Rotation In a rotation, you rotate a figure around a fixed point.
Geometry in Motion
Rotation
In a rotation, you rotate a figure around a fixed point.
Rotations are sometimes called _____.
turns
30°

15. This one-to-one correspondence is an example of a _______. mapping
Geometry in Motion A B C D E F
Each point on
the original
figure is called
a _________.
Its matching
point on the
corresponding
figure is called
its ______.
preimage
image
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.
This one-to-one correspondence is an example of a _______.
mapping

16. The symbol → is used to indicate a mapping.
Geometry in Motion A B C D E F
Each point on
the original
figure is called
a _________.
Its matching
point on the
corresponding
figure is called
its ______.
preimage
image
The symbol → 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.

17. → → → → → → A D AB DE B E BC EF C F CA FD Each point on Its matching
Geometry in Motion A B C D E F
Each point on
the original
figure is called
a _________.
Its matching
point on the
corresponding
figure is called
its ______.
preimage
image
Preimage
Image
Preimage
Image → A D → AB DE → B E BC → EF → C F → CA FD
This mapping is called a _____________.
transformation

18. Translations, reflections, and rotations are all __________.
Geometry in Motion
Translations, reflections, and rotations are all __________.
isometries
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.
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19. ΔABC  ΔXYZ What You'll Learn
Congruent Triangles
What You'll Learn
You will learn to identify corresponding parts of congruent
triangles
If a triangle can be translated, rotated, or reflected onto another triangle, so
that all of the vertices correspond, the triangles are _________________.
congruent triangles
The parts of congruent triangles that “match” are called ______________.
congruent parts
The order of the ________ indicates the corresponding parts!
ΔABC  ΔXYZ
vertices

20. These relationships help define the congruent triangles.
In the figure, ΔABC 
ΔFDE. A
As in a mapping, the order of the _______ indicates the corresponding parts.
vertices C B
Congruent Angles
Congruent Sides
A  F
AB  FD F E D
B  D
BC  DE
C  E
AC  FE
These relationships help define the congruent triangles.

21. If the _________________ of two triangles are congruent, then
Congruent Triangles
Definition of
Congruent
Triangles
If the _________________ of two triangles are congruent, then
the two triangles are congruent.
corresponding parts
If two triangles are _________, then the corresponding parts
of the two triangles are congruent.
congruent

22. ΔRST  ΔXYZ ΔRST  ΔXYZ. Find the value of n.
Congruent Triangles
ΔRST  ΔXYZ. Find the value of n. T S R Z X Y
40°
(2n + 10)°
50°
90°
ΔRST  ΔXYZ
identify the corresponding parts
corresponding parts are congruent
S  Y
50 = 2n + 10
subtract 10 from both sides
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40 = 2n
divide both sides by 2
20 = n

23. SSS and SAS
What You'll Learn
You will learn to use the SSS and SAS tests for congruency.

24. 4) Construct a segment congruent to CB. 5) Label the intersection F.
SSS and SAS
4) Construct a segment congruent to CB.
5) Label the intersection F.
2) Construct a segment congruent to AC. Label the endpoints of the segment D and E.
1) Draw an acute scalene triangle on a piece of paper. Label its vertices
A, B, and C, on the interior of each angle.
6) Draw DF and EF.
3) Construct a segment congruent to AB. A C B D E F
This activity suggests the following postulate.

25. Triangles are congruent. sides three corresponding
SSS and SAS
Postulate 5-1
SSS
Postulate
If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two
Triangles are congruent.
sides
three
corresponding A B C R S T
If AC 
RT and
AB 
RS and
BC  ST
then ΔABC
 ΔRST

26. In two triangles, ZY  FE, XY  DE, and XZ  DF.
SSS and SAS
In two triangles, ZY  FE, XY  DE, and XZ  DF.
Write a congruence statement for the two triangles. X D Z Y F E
Sample Answer:
ΔZXY  ΔFDE

27. In a triangle, the angle formed by two given sides is called the
SSS and SAS
In a triangle, the angle formed by two given sides is called the
____________ of the sides.
included angle
C is the included
angle of CA and CB A B C
A is the included
angle of AB and AC
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
by using two sides and the ____________.
included angle

28. If ________ and the ____________ of one triangle are
SSS and SAS
Postulate 5-2
SAS
Postulate
If ________ and the ____________ of one triangle are
congruent to the corresponding sides and included angle of
another triangle, then the triangles are congruent.
two sides
included angle A B C R S T
If AC 
RT and
A 
R and
AB  RS
then ΔABC
 ΔRST

29. NO! On a piece of paper, write your response to the following:
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. P R Q F E D
NO!
D is not the included angle for DF and EF.

30. ASA and AAS
What You'll Learn
You will learn to use the ASA and AAS tests for congruency.

31. It is the one side common to both angles.
ASA and AAS
The side of a triangle that falls between two given angles is called the ___________ of the angles.
included side
It is the one side common to both angles. A B C
AC is the included
side of A and C
CB is the included
side of C and B
AB is the included
side of A and B
You can show that two triangles are congruent by using _________ and the
___________ of the triangles.
two angles
included side

32. If _________ and the ___________ of one triangle are
ASA and AAS
Postulate 5-3
ASA
Postulate
If _________ and the ___________ of one triangle are
congruent to the corresponding angles and included side of
another triangle, then the triangles are congruent.
two angles
included side A B C R S T
If A 
R and
AC 
RT and
C  T
then ΔABC
 ΔRST

33. CA and CB are the nonincluded
ASA and AAS
CA and CB are the nonincluded
sides of A and B A B C
You can show that two triangles are congruent by using _________ and a
______________.
two angles
nonincluded side

34. If _________ and a ______________ of one triangle are
ASA and AAS
Theorem 5-4
AAS
Theorem
If _________ and a ______________ of one triangle are
congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent.
two angles
nonincluded side A B C R S T
If A 
R and
C 
T and
CB  TS
then ΔABC
 ΔRST

35. ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to
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 F E L M N

36. ASA and AAS