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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. - PowerPoint PPT Presentation
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Lesson 4.1Classifying Triangles
Today, you will learn to…* classify triangles by their sides and
angles* find measures in triangles
A
BC
ABC
Equilateral Triangle
3 congruent sides
Isosceles Triangle
2 congruent sides
Scalene Triangle
no congruent sides
Equiangular Triangle
3 congruent angles
Acute Triangle
3 acute angles60°
70°
50°
Obtuse Triangle
1 obtuse angle
95°
25°60°
Right Triangle
1 right angle
60°
30°
We classify triangles by their sides and angles.
SIDES ANGLESEquilateralIsoscelesScalene
EquiangularAcuteObtuseRight
A
BC
_____ is opposite A.CB
_____ is opposite B.AC
_____ is opposite C.AB
Identify the side opposite the given angle.
hypotenuse
leg
leg?
Leg?
base
leg
leg
Leg?
?
Theorem 4.1
Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is
________180°
1. Find m X.
61º
75º
Y mX =
Z X
44˚
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.
54˚
F
D
E
mF =
54˚ + mF = 90˚
36˚
3. Find m 1 and m 2.
2
50º
70º
exterior angle
adjacent angles1
m1 =m2 =60˚
120˚
Theorem 4.2Exterior 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.
4
1
2 3
m1 + m2 = m4
4
1
2 3
m1 + m2 + m3 = 180˚ m4 + m3 = 180˚
m1 + m2m4
m1 + m2 = m4 Sum of
nonadjacent interior s
= ext.
E
60˚F
4. Find mE.
m E =
110˚D G
m E + 60˚ = 110˚
50˚
E
60˚F
5. Find x.
x =
(3x + 10)˚D G
x + 60 = 3x + 10
25x˚
Lesson 4.2Congruence and
Triangles
Today, you will learn to…* identify congruent figures and corresponding parts* prove that 2 triangles are congruent
Figures are congruent if and only if all pairs of
corresponding angles and sides are congruent.
Def. of Congruent Figures
Statement of CongruenceΔ ABC Δ XYZ vertices are written in corresponding order
AB
BC
AC
XY
YZ
XZ
A
B
C
X
Z
Y
XZ
YZ
XY
E
F D
B
A C
1. Mark ΔDEF to show that Δ ABC Δ DEF.
2. Find all missing measures.
ABC DEF
B E
F
DB
A C
35˚
108.2 5.7
8.2
10
5.755˚
35˚
55˚
?
?
?
?
?
?
?
3. In the diagram, ABCD KJHL. Find x and y.
A
C
L
K
J
H
B
D9 cm
(4x – 3) cm(3y)˚
85˚
93˚
75˚
x = 3 y = 254x-3 = 3y =9 75˚
4. ΔABC ΔDEF. Find x.A
CB
93˚30˚
F
(4x + 15)˚ D
Ex = 10.5
4x + 15 = 57
57
70˚
Theorem 4.3
Third Angles TheoremIf 2 angles of one triangle are
congruent to 2 angles of another triangle, then…
the third angles are also
congruent.
D
O GC
A T
70˚
60˚
60˚
?
?
E
F
G
J
H
58°
58°
5. Decide if the triangles are congruent. Justify your reasoning.
ΔEFG Δ______J H G
Vertical Angles Theorem
Third Angles Theorem
W X
YZ
M
6.
1) WX YZ , WX | | YZ,M is the midpoint of WY and XZ2) 3)4)5) ΔWXM ΔYZM
1 2
3
4
5 6
1 6 2) Alt. Int. s Theorem
1) Given
3 4 3) Vertical Angles Th.WM MY and ZM MX4) Def. of midpoint
5) Def. of figures
and 2 5
7. Identify any figures you can prove congruent & write a congruence statement.
A B
CD
Reflexive Property Alt. Int. Th.
Third Angle Th.ACD C AB
Theorem 4.4
Properties of Congruent Triangles
Reflexive
Symmetric
Transitive
ABC ABC If ABC XYZ,
then XYZ ABCIf ABC XYZ
and XYZ MNO then ABC
MNO
Lesson 4.3Proving Triangles
are CongruentToday, 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
X
Y
Z
A C
B
If Side AB XY Side AC XZ Side BC YZ,then ΔABC ΔXYZ
by SSS
If 3 pairs of sides are congruent, then the two triangles are congruent.
1. Does the diagram give enough info to use SSS Congruence?
A
B
C
J
K L
no
Given: LN NP and M is the midpoint of LPProve: ΔNLM ΔNPM
N
LM
P
2.
Def of midpointLM MPReflexive PropertyNM 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).
N
P
MD
F
E
NM = MP = NP =
DE = EF = DF =
45
45
41
41(- 5 – - 1)2 + (1 – 6)2(6 – 2)2 + (1 – 6)2
Using 2 congruent segments and 1 included angle, can you ONLY
create 2 triangles that are congruent?
SAS Experiment
Side-Angle-Side Congruence Postulate
If Side AB XY Angle B Y Side BC YZ,then ΔABC ΔXYZ
by SAS X
Y
Z
A C
B
If 2 pairs of sides and their included angle are congruent, then the two
triangles are congruent.
SAS?
4. 5.
6. 7.
SAS
SAS
NO!
NO!
ABD _ _ _ by SAS
8. Does the diagram give enough info to use SAS Congruence?
A
B CDACD
9. Does the diagram give enough info to use SAS Congruence?
V W
X
ZY
no
10. Does the diagram give enough info to use SAS Congruence?
A E
C DB
no
Given: W is the midpoint of VY and the midpoint of ZXProve: ΔVWZ ΔYWX
11.
X
Z
W
Y
V
VW WY and ZW WX Def. of midpointVWZ YWX Vertical Angles ThVWZ YWX SAS Congruence
12.Given: AB PB , MB APProve: ΔMBA ΔMBP
M
PBA
MB MB Reflexive Property
ABM & PBM are right s Def of lines
MBA MBP SAS Congruence
ABM PBM All right s are
What is the best way to get better at proofs?
Lesson 4.4Proving Triangles
are CongruentToday, you will learn to…* prove that triangles are congruent* use congruence postulates to solve
problems
Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent?
ASA Experiment
Angle-Side-Angle Congruence Postulate
If Angle B Y, Side BC YZ, Angle C Zthen ΔABC ΔXYZ
by ASA X
Y
Z
A C
B
If 2 pairs of angles and the included sides are congruent, then the two
triangles are congruent.
A
B C
Included side?
The included side between
A and B is _____AB
The included side between
B and C is _____
A
B C
CB
Included side?
A
B C
ACThe included side between
A and C is _____
Included side?
ASA?
1. 2.
3. 4.
ASA
ASA
NO!
NO!
5. Does the diagram give enough info to use ASA Congruence?
AB C
DΔ ABD Δ by ASA
Third Angles Theorem
Reflexive Property
ACD
6. Does the diagram give enough info to use ASA Congruence?
A
B C
D
yes, Δ ACB ______ by ASA
Δ CAD
Alt. Int. Angles Theorem
Reflexive Property
7. Does the diagram give enough info to use ASA Congruence?
A
B C
D
no
Reflexive Property
8. Does the diagram give enough info to use ASA Congruence?
A
B
C
J
K L
KLJ _ _ _ by ASA ACB
9. Determine whether the triangles are congruent by ASA.
L K
GH
J
HJG _ _ _ by ASA
Vertical Angles Theorem
Alt. Int. Angles Theorem
KJL
Angle-Angle-Side Congruence Theorem
If Angle B Y Angle C Z Side AB XYthen ΔABC ΔXYZ
by AAS X
Y
Z
A C
B
If 2 pairs of angles and a pair of nonincluded sides are congruent, then
the two triangles are congruent.
AAS?
10. 11.
12. 13.
NO!
NO!
AAS
AAS
14. Does the diagram give enough info to use AAS Congruence?
AB C
D
ABD _ _ _ by AAS ACDACDACD
Reflexive Property
15. Does the diagram give enough info to use AAS Congruence?
A
B
C
J
K L
KLJ _ _ _ by AAS ACBACBACB
16. Determine whether the triangles are congruent by AAS.
L K
GH
J
HJG _ _ _ by AAS
Vertical AnglesTheorem
Alt. Int. AnglesTheorem
K JL
SSA Experiment
Using 2 sides and 1 angle that is NOT included, can you ONLY create two
triangles that are congruent?
NO
AAA ExperimentUsing 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 AAASSA SASASA 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
C
A
B
X
YZ
C Z
Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ?18. CB ZY , AC
XZSSS Congruence
C
A
B
X
YZ
AB XY
Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ?19. CB ZY , C
ZSAA Congruence
C
A
B
X
YZ
A X
What is the best way to get better at proofs?
Lesson 4.5Corresponding Parts of Congruent Triangles are
CongruentToday, you will learn to…* use congruence postulates to solve
problems CPCTC
1. Given: AB || CD , BC || DA Prove: AB CD B C
DA1 2 Alt. Int. Angles Theorem, 3 4
1
2
3
4
Reflexive Property BD BD ASA ABD CDB
AB CD CPCTC
C A
D
B
12
43
2. Given: 1 2 , 3 4 Prove: CD CB
Reflexive Property CA CA ASA ABC ADC
CD CB CPCTC
A
B
DC
3. Given: AC AD , BC BD Prove: C D
C D
Reflexive Property AB AB SSS ABC ABDCPCTC
4. Given: A is the midpoint of MTA is the midpoint of SR Prove: MS || TR
M
S
A
T
R
Def. of midpointVertical Angles Theorem
MA AT
SAS SAM R AT
MS | | TR CPCTC
SAM RATSA AR
Alt. Int. Angles ConverseM T
Triangle Congruence?
SASSSA
AASASA
SSS AAA
2 angles & 1 side?
2 sides & 1 angle?
3 sides or 3 angles?
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 4.6Isosceles, Equilateral, and Right Triangles
Today, you will learn to…* use properties of isosceles, equilateral, and right triangles
Students need rulers and protractors.
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?
base angles
legleg
base
Theorem 4.6
Base Angles Theorem
If 2 sides of a triangle are congruent, then … the
angles opposite them are congruent.
Theorem 4.7
Base Angles ConverseIf 2 angles of a triangle are congruent, then the sides
opposite them are congruent.
B
CA D
A C by the
Base Angles Theorem
1. What angles are congruent?
B
CA D
AB BC
2. What sides are congruent?
by the Base Angles
Converse
3. Find mB.
A B
C
75˚
mB = 75˚
4. Find mB.
A B
C
68˚
mB =68˚
44˚?
5. Find x.
A B
C
2x + 4
2x + 4 = 18
18
x = 7
2x = 14
6. Find x.
A
B
C
6x - 10
6x – 10 = 55
6x = 15
x = 2.5
4
x˚
7. Find x and y.
50˚
y˚
x =y =
115˚ x˚65˚
y˚
6532.5
??
?
Corollaries to Theorem 4.6/4.7(hint: don’t write these yet)
If a triangle is equilateral, then it is equiangular.
ANDIf a triangle is equiangular,
then it is equilateral.
A triangle is equilateral if and only if
it is equiangular.
Corollaries to Theorem 4.6/4.7
8. Find x.
A B
C
247x + 3 = 24
7x + 3x = 3
7x = 21 10x
– 6
10x – 6 = 7x + 3
3x = 910x - 6 = 24
10x = 30
9. Find x.
A
B
CWhat is the measure of each angle?
2x =
x = 30
2x˚
60˚
x˚
10. Find x and y.
50˚
y˚
x =y =
60˚70˚
70˚
50˚
80 40
? ??
?
60˚
60˚
ExperimentUsing a right angle,
a hypotenuse, and a leg, can you ONLY create 2 triangles
that are congruent?
Hypotenuse-Leg Congruence Theorem
X
Y
Z
A C
B
The triangles MUST be right triangles.
If Hyp BC YZ Leg AB XYthen ΔABC ΔXYZ
by HL
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?
W
X
Z
Y
NO
Reflexive Property
12. X is a midpoint. Does the diagram give enough info to use HL?
VWX _ _ _ by HL
V W
X
ZYYZX
Def. of midpoint
13. Does the diagram give enough info to use HL Congruence? W
X
Z
Y
YWX _ _ _ by HL YZX
Base Angles Converse
Reflexive Property