Download Lesson 4.1 Classifying Triangles

Lesson 4.1
Classifying
Triangles
Today, you will learn to…
* classify triangles by their sides and
angles
* find measures in triangles
 ABC
A
B
C
Equilateral Triangle
3 congruent sides
Isosceles Triangle
2 congruent sides
Scalene Triangle
no congruent sides
Equiangular Triangle
3 congruent angles
Acute Triangle
70°
60°
50°
3 acute angles
Obtuse Triangle
95°
60°
25°
1 obtuse angle
Right Triangle
30°
60°
1 right angle
We classify triangles by their
sides and angles.
SIDES
Equilateral
Isosceles
Scalene
ANGLES
Equiangular
Acute
Obtuse
Right
Identify the side
opposite the
given angle.
A
C
B
CB is opposite
_____
A.
AC is opposite
_____
B.
AB is opposite
_____
C.
Leg?
leg
hypotenuse
?
leg
Leg?
leg
? base
leg
Theorem 4.1
Triangle Sum Theorem
The sum of the measures of the
interior angles of a triangle is
________
180°
1. Find m  X.
Y
mX = 44˚
61º
Z
75º
X
If the sum of the interior
angles is 180º, what
do you know about
1 and 2?
1
2
Corollary to the Triangle
Sum Theorem
The acute angles of a
right triangle are
complementary
_________________.
2. Find m F.
D
54˚ + mF = 90˚
54˚
E
mF = 36˚
F
3. Find m  1 and m  2.
50º
70º
m1 = 60˚
m2 = 120˚
exterior
angle
1 2
adjacent angles
Theorem 4.2
Exterior Angle Theorem
The measure of an exterior angle
of a triangle is equal to the sum of
the measures of the 2 nonadjacent
interior angles.
1
2
m1 + m2 = m4
3 4
m1 + m2 + m3 = 180˚
m4 + m3 = 180˚
m1 + m2 = m4
1
2
Sum of
nonadjacent
interior s
3 4
= ext. 
4. Find mE.
E
m E + 60˚ = 110˚
m E = 50˚
60˚
D
110˚
F
G
5. Find x.
E
x˚
60˚
D
x + 60 = 3x + 10
x = 25
(3x + 10)˚
F
G
Lesson 4.2
Congruence and
Triangles
Today, you will learn to…
* identify congruent figures and
corresponding parts
* prove that 2 triangles are congruent
Def. of Congruent Figures
Figures are congruent
if and only if all pairs of
corresponding angles and
sides are congruent.
Statement of Congruence
Δ ABC  Δ XYZ
vertices are written in
corresponding order
AX
BY
CZ
XY
AB  XY
BC  YZ
AC  XZ
XZ
1. Mark ΔDEF to show that
Δ ABC  Δ DEF.
B
C
A
F
E
D
2. Find all missing measures.
ABC  DEF
B
B
?
8.2
A
D
10
?
8.2
?
?
5.7
?
55˚
C
5.7
?
?
55˚
F
E
35˚
10
3. In the diagram, ABCD  KJHL.
Find x and y.
K
A
B
93˚
85˚
D
9 cm
75˚
J
L
(3y)˚
(4x – 3) cm
H
C 4x-3 = 9
x=3
3y = 75˚
y = 25
4. ΔABC  ΔDEF. Find x.
A
93˚
57
30˚
4x + 15 = 57
x = 10.5
B
C
F
(4x + 15)˚
D
E
Theorem 4.3
Third Angles Theorem
If 2 angles of one triangle are
congruent to 2 angles of another
D
triangle, then…
O
60˚
70˚ ?
C
G
A
60˚
70˚ ?
the third angles
are also
congruent.
T
5. Decide if the triangles are
congruent. Justify your reasoning.
Vertical Angles Theorem H
E
58°
G
58°
F
J
Third
Angles
Theorem
HG
ΔEFG  Δ______
J
6.
W
1
2
M
Z
5
X
3
4
6
Y
1) WX  YZ , WX | | YZ, 1) Given
M is the midpoint of WY and XZ
2) 1  6 and 2  5 2) Alt. Int. s Theorem
3) 3  4
3) Vertical Angles Th.
4) WM  MY and ZM  MX 4) Def. of midpoint
5) Def. of  figures
5) ΔWXM  ΔYZM
7. Identify any figures you can prove
congruent & write a congruence
statement.
B
A
D
C
Alt.
Int.

Th.
Reflexive
Property
ACD   C AB
Third Angle Th.
Theorem 4.4
Properties of Congruent
Triangles
ABC  ABC
Reflexive
If ABC  XYZ,
Symmetric
then XYZ  ABC
If ABC  XYZ
Transitive
and XYZ  MNO
then ABC  MNO
Lesson 4.3
Proving Triangles
are Congruent
Today, you will learn to…
* prove that triangles are congruent
* use congruence postulates to solve
problems
SSS Experiment
Using 3 segments, can you ONLY
create 2 triangles that are
congruent?
Side-Side-Side
Congruence Postulate
B
If Side AB  XY
C
A
Y
Side AC  XZ
Side BC  YZ,
then ΔABC  ΔXYZ
by SSS
X
Z
If 3 pairs of sides are congruent, then
the two triangles are congruent.
1. Does the diagram give enough
info to use SSS Congruence?
K
L
C
A
no
J
B
2. Given: LN  NP and
M is the midpoint of LP
Prove: ΔNLM  ΔNPM
N
L
P
M
Def of midpoint
Reflexive Property
LM  MP
NM  NM
NLM  NPM SSS Congruence
3. Show that ΔNPM  ΔDFE by
SSS if N(-5,1), P (-1,6), M (-1,1),
D (6,1), F (2,6), and E (2,1).
F
P
N
D
M
E
(- (6
5 –– -2)
1)22++(1
(1––6)
6)22
NM = 4
MP = 5
NP = 41
DE = 4
EF = 5
DF = 41
SAS Experiment
Using 2 congruent segments and
1 included angle, can you ONLY
create 2 triangles that are
congruent?
B
Side-Angle-Side
Congruence Postulate
If Side AB  XY
C
A
Y
Angle B  Y
Side BC  YZ,
then ΔABC  ΔXYZ
by SAS
X
Z
If 2 pairs of sides and their included
angle are congruent, then the two
triangles are congruent.
SAS?
5.
4.
NO!
SAS
6.
7.
NO!
SAS
8. Does the diagram give enough
info to use SAS Congruence?
A
B
D
C
ABD  AC
_ _D
_ by SAS
9. Does the diagram give enough
info to use SAS Congruence?
W
V
no
Y
X
Z
10. Does the diagram give enough
info to use SAS Congruence?
E
A
no
B
C
D
11. Given:
W is the midpoint of VY
and the midpoint of ZX
Prove: ΔVWZ  ΔYWX
VW  WY and ZW  WX
Def. of midpoint
VWZ YWX Vertical Angles Th
VWZ  YWX SAS Congruence
12. Given: AB  PB , MB  AP
Prove: ΔMBA  ΔMBP
M
A
B
P
ABM & PBM are right s Def of  lines
ABM  PBM
MB  MB
MBA  MBP
All right s are 
Reflexive Property
SAS Congruence
What is the best way to
get better at proofs?
Lesson 4.4
Proving Triangles
are Congruent
Today, you will learn to…
* prove that triangles are congruent
* use congruence postulates to solve
problems
ASA Experiment
Using 2 angles connected by 1
segment, can you ONLY create
two triangles that are congruent?
Angle-Side-Angle
Congruence Postulate
B
If Angle B  Y,
C
A Y
Side BC  YZ,
Angle C  Z
then ΔABC  ΔXYZ
X
Z
by ASA
If 2 pairs of angles and the included
sides are congruent, then the two
triangles are congruent.
Included side?
A
B
C
The included side between
AB
 A and  B is _____
Included side?
A
B
C
The included side between
CB
 B and  C is _____
Included side?
A
B
C
The included side between
 A and  C is _____
AC
ASA?
2.
1.
NO!
ASA
3.
4.
NO!
ASA
5. Does the diagram give enough
info to use ASA Congruence?
B
A
C
Reflexive Property
Third Angles Theorem
D
Δ ABD  Δ ACD by ASA
6. Does the diagram give enough
info to use ASA Congruence?
A
D
Reflexive
Property
B
Alt. Int.
Angles
Theorem
C
yes, ΔACB  ______
ΔC A D by ASA
7. Does the diagram give enough
info to use ASA Congruence?
A
D
Reflexive
Property
B
C
no
8. Does the diagram give enough
info to use ASA Congruence?
K
L
J
C
B
A
KLJ  ACB
_ _ _ by ASA
9. Determine whether the triangles
are congruent by ASA.
L
K
Vertical Angles
Theorem
H
J
Alt. Int. Angles
Theorem
G
HJG   K
_ _J L_ by ASA
Angle-Angle-Side
Congruence Theorem
B
If Angle B  Y
C
A
Angle C  Z
Y
Side AB  XY
then ΔABC  ΔXYZ
by AAS
Z
X
If 2 pairs of angles and a pair of
nonincluded sides are congruent, then
the two triangles are congruent.
AAS?
11.
10.
AAS
NO!
12.
13.
AAS
NO!
14. Does the diagram give enough
info to use AAS Congruence?
B
A
C
Reflexive
Property
D
ABD   ACD
_ _ _ by AAS
15. Does the diagram give enough
info to use AAS Congruence?
K
L
J
C
A
B
KLJ  ACB
_ _ _ by AAS
16. Determine whether the triangles
are congruent by AAS.
L
K
Vertical Angles
Theorem
H
J
Alt. Int. Angles
Theorem
G
HJG  K
_ _JL_ by AAS
SSA Experiment
Using 2 sides and 1 angle that is NOT
included, can you ONLY create two
triangles that are congruent?
NO
AAA Experiment
Using 3 angles, can you ONLY create
two triangles that are congruent?
NO
All of the angles are ,
but the s are NOT 
Triangle
Congruence?
SSS
SSA
ASA
AAA
SAS
AAS
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
17. CB  ZY , AC  XZ
SAS Congruence
X
A
C
B
Z
C  Z
Y
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
18. CB  ZY , AC  XZ
SSS Congruence
X
A
C
B
Z
AB  XY
Y
Mark the given information on the
triangles. What additional congruence
would you need to show ABC  XYZ?
19. CB  ZY , C  Z
SAA Congruence
X
A
C
B
Z
A  X
Y
What is the best way to
get better at proofs?
Lesson 4.5
Corresponding Parts of
Congruent Triangles are
Congruent
Today, you will learn to…
* use congruence postulates to solve
problems
CPCTC
1. Given: AB || CD , BC || DA
Prove: AB  CD B
3
1
A
1  2 , 3  4
BD  BD
ABD   C D B
AB  CD
C
4
2
D
Alt. Int. Angles Theorem
Reflexive Property
ASA
CPCTC
2. Given: 1  2 , 3  4
D
Prove: CD  CB
C
CA  CA
ABC   A D C
CD  CB
1
2
4
3
B
Reflexive Property
ASA
CPCTC
A
3. Given: AC  AD , BC  BD A
Prove: C  D
C
B
AB  AB
ABC   AB D
CD
Reflexive Property
SSS
CPCTC
4. Given:
A is the midpoint of MT M
A is the midpoint of SR
R
A
Prove: MS || TR
S
T
Def. of midpoint
MA  AT SA  AR
Vertical Angles Theorem
SAM  RAT
SAS
SAM   R AT
CPCTC
M  T
Alt. Int. Angles Converse
MS | | TR
Triangle Congruence?
2 angles
& 1 side?
2 sides &
1 angle?
AAS
ASA
SAS
SSA
3 sides or 3 angles?
SSS
AAA
You can ONLY use CPCTC after
you use one of these!
Does the quilt design have
vertical, horizontal, or diagonal
symmetry?
Does the quilt design have
vertical, horizontal, or diagonal
symmetry?
Lesson 4.6
Isosceles, Equilateral,
and Right Triangles
Students need rulers and protractors.
Today, you will learn to…
* use properties of isosceles,
equilateral, and right triangles
Use a ruler to draw two
congruent segments that
share one endpoint.
Connect the endpoints to
create a triangle.
Measure each interior
angle. What do you notice?
leg
leg
base
base angles
Theorem 4.6
Base Angles Theorem
If 2 sides of a triangle are
the
congruent, then …
angles opposite them
are congruent.
Theorem 4.7
Base Angles Converse
If 2 angles of a triangle are
congruent, then the sides
opposite them are congruent.
1. What angles
are congruent?
A  C
by the
Base Angles
Theorem A
B
D
C
2. What sides
are congruent?
B
AB  BC
by the
Base Angles A
Converse
D
C
3. Find mB.
C
mB = 75˚
A
75˚
B
4. Find mB.
C
mB = 44˚
A
68˚
?
68˚
B
5. Find x.
C
2x + 4 = 18
2x = 14
18
2x + 4
x=7
A
B
6. Find x.
C
6x – 10 = 5
B
6x = 15
x = 2.5
A
7. Find x and y.
y˚
? 50˚
y˚
y = 32.5
115˚
? 65˚
x˚
x˚
?
x = 65
Corollaries to Theorem 4.6/4.7
(hint: don’t write these yet)
IfAatriangle
triangle is equilateral,
equilateral
thenifit and
is equiangular.
only if
AND
it is equiangular.
If a triangle is equiangular,
then it is equilateral.
8. Find x.
7x + 3 = 24
7x = 21
x=3
C
24
A
10x – 6 = 7x + 3
3x = 9
7x + 3
B
10x - 6 = 24
10x = 30
9. Find x.
C
What is the
measure of
each angle?
2x˚
2x = 60˚
x = 30
B
A
10. Find x and y.
y˚
50˚
?
y = 80
x˚
70˚
?
60˚
70˚
? 60˚ 60˚
?
50˚
x = 40
Experiment
Using a right angle,
a hypotenuse, and a leg,
can you ONLY create 2 triangles
that are congruent?
Hypotenuse-Leg
Congruence Theorem
Y
If Hyp BC  YZ
Leg AB  XY
then ΔABC  ΔXYZ
X
by HL
B
A
C
Z
The triangles MUST
be right triangles.
If the hypotenuse and a leg of two right
triangles are congruent, then the two
triangles are congruent.
11. Does the diagram give enough
info to use HL Congruence?
Reflexive Property
W
X
Y
NO
Z
12. X is a midpoint. Does the diagram
give enough info to use HL?
V
Def. of midpoint
Y
W
X
Z
VWX  YZ
_ _X
_ by HL
13. Does the diagram give enough
info to use HL Congruence?
W
Reflexive Property
Y
X
Base Angles Converse
Z
YWX   YZ
_ _X
_ by HL