Transcript Chapter 4 Notes

Chapter 4: Congruent
Triangles
Lesson 1: Classifying Triangles
Classifying Triangle by Angles
 Acute Triangle: all of the angles are acute
 Obtuse Triangle: one angle is obtuse, the
other two are acute
 Right Triangle: one angle is right, the other
two are acute
 Equiangular Triangle: all the angles are 60
degrees
Classifying Triangles by Sides
 Scalene Triangle: all sides are
7
different measures
3
5
 Isosceles Triangle: at least two sides
* vertex angle= formed by the two
sides of an isosceles
have the same measure congruent
triangle
* base= the side of an isosceles
triangle not congruent to the others
 Equilateral Triangle: all sides have the
same measure
If point Y is the midpoint of VX,
and WY = 3.0 units, classify
ΔVWY as equilateral, isosceles,
or scalene. Explain your
reasoning.
ALGEBRA Find the measure of the
sides __
of isosceles triangle KLM
with base KL.
ALGEBRA Find x and the measure of each side of
equilateral triangle ABC if AB = 6x – 8, BC = 7 + x,
and AC = 13 – x.
 Find the measure of each side of
Triangle JKL and classify the triangle
based on its sides.
 J(-3, 2) K(2, 1) L(-2, -3)
Find
y___
___
Chapter 4: Congruent
Triangles
Lesson 2: Angles of Triangles
 The sum of the
measures of the
angles of a triangle
is always 180
degrees.
 The acute angles of
a right triangle are
complementary
 There can be at
most one right or
one obtuse angle in
a triangle
 Third Angle Theorem
 If two angles of one
triangle are
congruent to two
angles of another
triangle, then the
third angles of the
triangles are also
congruent. X
A
B
Y
Z
C
If  A   X, and B   Y, then
 C   Z.
Interior and Exterior Angles of
Triangles
 Exterior angle: formed by one side of
a triangle and the extension of
another side
 The interior angles farthest from the
exterior angle are its remote interior
angles. (remote interior angles are not adjacent to the
exterior angle)
Remote interior
angles
Exterior
angle
2
An exterior angle is equal to the
sum of its remote interior angles.
ex:
1
3
4
1+

2=

4

Anticipation Guide: read each statement. State whether the
sentence is true or false. If the statement is false- rewrite it with
the correct term in place of the underlined word
 The acute angles of a right triangle are
supplementary
 The sum of the measures of the angles of any
triangle is 100
 A triangle can have at most one right angle or
acute angle
 If two angles of one triangle are congruent to two
angles of another triangle, then the third angle of
the triangles are congruent
 The measure of an exterior angle of a triangle is
equal to the difference of the measures of the two
remote interior angles
 If the measures of two angles of a triangle are 62
and 93, then the measure of the third angle is 35
 An exterior angle of a triangle forms a linear pair
with an interior angle of the triangle
SOFTBALL The diagram
shows the path of the
softball in a drill
developed by four players.
Find the measure of each
numbered angle.
Find the measure of each
numbered angle.
GARDENING Find the measure
of FLW in the fenced flower
garden shown.
The piece of quilt fabric is in the shape of a
right triangle. Find the measure of ACD.
Find the measure of
each numbered
angle.
Find m3.
Chapter 4: Congruent
Triangles
Lesson 6: Isosceles Triangles
Isosceles Triangles
.
Vertex Angle
leg
- If two sides of a triangle are
congruent, the two angles
opposite of them are also
congruent
leg
-If two angles of a triangle
are congruent, then two
sides opposite of them are
also congruent
Base angles
- If a triangle is equilateral, it
is also equiangular
A. Find mR.
B. Find PR
A. Find mT.
ALGEBRA Find the value of each variable
Chapter 4: Congruent
Triangles
Lesson 3: Congruent Triangles
Definition of Congruent
Triangles
 Congruent triangles are triangles with
exactly the same size and shape
 CPCTC: Corresponding Parts of
Congruent Triangles are Congruent
 Two triangles are congruent if and only if
their corresponding parts are congruent
Corresponding Parts
A
 Corresponding
parts have the
same congruence
markings

B
C
H
I
J
 AB  HI
 AC  HJ
 BC  IJ
 A   H
 B   I
 C   J
Congruence Transformations
 Slide or Translation: the triangle is
in the same position farther down,
up, or across the page
 Turn or Rotation: the triangle is
spun around a point (usually one of
the angles)
 Flip or reflection: the triangle is
shown in a mirror image across a line
of symmetry
Write a congruence
statement for the
triangles.
Name the
corresponding
congruent angles for
the congruent
triangles.
In the diagram, ΔITP  ΔNGO. Find the values of
x and y.
In the diagram, ΔFHJ  ΔHFG. Find the values of
x and y.
Find the missing information in the following proof.
Prove: ΔQNP  ΔOPN
Proof:
1. Given
2. Reflexive Property of
2.
Congruence
3. Q  O, NPQ  PNO 3. Given
4. _________________
4. QNP  ONP
?
1.
5. ΔQNP  ΔOPN
5. Definition of Congruent
Polygons
Write a two-column proof.
Prove: ΔLMN  ΔPON
Chapter 4: Congruent
Triangles
Lesson 4 and 5: Proving CongruenceSSS, SAS, ASA, AAS, and HL
SSS
 Side-Side-Side
 If all three sets of corresponding sides are
congruent, the triangles are congruent
A
M
C
ABC

MNO
N
O
SAS
 Side-Angle-Side
 If two corresponding sides and the included
angles of two triangles are congruent, then the
triangles are congruent
* The included
angle is the angle
X
F
between the
congruent sides
Y
Z
XYZ

G
FGH
H
ASA
 Angle-Side-Angle
 If two sets of corresponding angles and the
included sides are congruent, then the triangles
are congruent
* The included side is
the side between the
J
R
two congruent angles
L
K
JKL

RST
T
S
AAS
 Angle-Angle-Side
 If two sets of corresponding angles and one of
the corresponding non-included sides are
congruent, then the triangles are congruent
T
E
G
F
EFG

V
TUV
U
HL
 Hypotenuse-Leg
 If the hypotenuse and one set of corresponding legs
of two right triangles are congruent, then the
triangles are congruent
C
R
D
H
CDH

RAM
A
M
Determine if the triangles are
congruent. If they are, write the
congruence statement.
___ ___
Given: AC  AB
D is the midpoint of BC.
Prove: ΔADC  ΔADB
Determine whether ΔABC  ΔDEF
for A(–5, 5), B(0, 3), C(–4, 1),
D(6, –3), E(1, –1), and F(5, 1).
Determine if the triangles are
congruent. If they are, write the
congruence statement.
Determine which
postulate can be used
to prove that the
triangles are
congruent. If it is not
possible to prove
congruence, choose
not possible.
Write a two column proof.