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Lesson 4.1 Classifying Triangles

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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 4.1 Classifying Triangles Today, you will learn to… * classify triangles by their sides and angles * find measures in triangles
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Page 1: Lesson 4.1 Classifying Triangles

Lesson 4.1Classifying Triangles

Today, you will learn to…* classify triangles by their sides and

angles* find measures in triangles

Page 2: Lesson 4.1 Classifying Triangles

A

BC

ABC

Page 3: Lesson 4.1 Classifying Triangles

Equilateral Triangle

3 congruent sides

Page 4: Lesson 4.1 Classifying Triangles

Isosceles Triangle

2 congruent sides

Page 5: Lesson 4.1 Classifying Triangles

Scalene Triangle

no congruent sides

Page 6: Lesson 4.1 Classifying Triangles

Equiangular Triangle

3 congruent angles

Page 7: Lesson 4.1 Classifying Triangles

Acute Triangle

3 acute angles60°

70°

50°

Page 8: Lesson 4.1 Classifying Triangles

Obtuse Triangle

1 obtuse angle

95°

25°60°

Page 9: Lesson 4.1 Classifying Triangles

Right Triangle

1 right angle

60°

30°

Page 10: Lesson 4.1 Classifying Triangles

We classify triangles by their sides and angles.

SIDES ANGLESEquilateralIsoscelesScalene

EquiangularAcuteObtuseRight

Page 11: Lesson 4.1 Classifying Triangles

A

BC

_____ is opposite A.CB

_____ is opposite B.AC

_____ is opposite C.AB

Identify the side opposite the given angle.

Page 12: Lesson 4.1 Classifying Triangles

hypotenuse

leg

leg?

Leg?

Page 13: Lesson 4.1 Classifying Triangles

base

leg

leg

Leg?

?

Page 14: Lesson 4.1 Classifying Triangles

Theorem 4.1

Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is

________180°

Page 15: Lesson 4.1 Classifying Triangles

1. Find m X.

61º

75º

Y mX =

Z X

44˚

Page 16: Lesson 4.1 Classifying Triangles

If the sum of the interior angles is 180º, what

do you know about 1 and 2?

1

2

Page 17: Lesson 4.1 Classifying Triangles

Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are

_________________.complementary

Page 18: Lesson 4.1 Classifying Triangles

2. Find m F.

54˚

F

D

E

mF =

54˚ + mF = 90˚

36˚

Page 19: Lesson 4.1 Classifying Triangles

3. Find m 1 and m 2.

2

50º

70º

exterior angle

adjacent angles1

m1 =m2 =60˚

120˚

Page 20: Lesson 4.1 Classifying Triangles

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

Page 21: Lesson 4.1 Classifying Triangles

4

1

2 3

m1 + m2 + m3 = 180˚ m4 + m3 = 180˚

m1 + m2m4

m1 + m2 = m4 Sum of

nonadjacent interior s

= ext.

Page 22: Lesson 4.1 Classifying Triangles

E

60˚F

4. Find mE.

m E =

110˚D G

m E + 60˚ = 110˚

50˚

Page 23: Lesson 4.1 Classifying Triangles

E

60˚F

5. Find x.

x =

(3x + 10)˚D G

x + 60 = 3x + 10

25x˚

Page 24: Lesson 4.1 Classifying Triangles

Lesson 4.2Congruence and

Triangles

Today, you will learn to…* identify congruent figures and corresponding parts* prove that 2 triangles are congruent

Page 25: Lesson 4.1 Classifying Triangles

Figures are congruent if and only if all pairs of

corresponding angles and sides are congruent.

Def. of Congruent Figures

Page 26: Lesson 4.1 Classifying Triangles

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

Page 27: Lesson 4.1 Classifying Triangles

E

F D

B

A C

1. Mark ΔDEF to show that Δ ABC Δ DEF.

Page 28: Lesson 4.1 Classifying Triangles

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˚

?

?

?

?

?

?

?

Page 29: Lesson 4.1 Classifying Triangles

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˚

Page 30: Lesson 4.1 Classifying Triangles

4. ΔABC ΔDEF. Find x.A

CB

93˚30˚

F

(4x + 15)˚ D

Ex = 10.5

4x + 15 = 57

57

Page 31: Lesson 4.1 Classifying Triangles

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˚

?

?

Page 32: Lesson 4.1 Classifying Triangles

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

Page 33: Lesson 4.1 Classifying Triangles

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

Page 34: Lesson 4.1 Classifying Triangles

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

Page 35: Lesson 4.1 Classifying Triangles

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

Page 36: Lesson 4.1 Classifying Triangles
Page 37: Lesson 4.1 Classifying Triangles

Lesson 4.3Proving Triangles

are CongruentToday, you will learn to…* prove that triangles are congruent* use congruence postulates to solve

problems

Page 38: Lesson 4.1 Classifying Triangles

SSS Experiment

Using 3 segments, can you ONLY create 2 triangles that are

congruent?

Page 39: Lesson 4.1 Classifying Triangles

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.

Page 40: Lesson 4.1 Classifying Triangles

1. Does the diagram give enough info to use SSS Congruence?

A

B

C

J

K L

no

Page 41: Lesson 4.1 Classifying Triangles

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

Page 42: Lesson 4.1 Classifying Triangles

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

Page 43: Lesson 4.1 Classifying Triangles

Using 2 congruent segments and 1 included angle, can you ONLY

create 2 triangles that are congruent?

SAS Experiment

Page 44: Lesson 4.1 Classifying Triangles

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.

Page 45: Lesson 4.1 Classifying Triangles

SAS?

4. 5.

6. 7.

SAS

SAS

NO!

NO!

Page 46: Lesson 4.1 Classifying Triangles

ABD _ _ _ by SAS

8. Does the diagram give enough info to use SAS Congruence?

A

B CDACD

Page 47: Lesson 4.1 Classifying Triangles

9. Does the diagram give enough info to use SAS Congruence?

V W

X

ZY

no

Page 48: Lesson 4.1 Classifying Triangles

10. Does the diagram give enough info to use SAS Congruence?

A E

C DB

no

Page 49: Lesson 4.1 Classifying Triangles

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

Page 50: Lesson 4.1 Classifying Triangles

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

Page 51: Lesson 4.1 Classifying Triangles

What is the best way to get better at proofs?

Page 52: Lesson 4.1 Classifying Triangles
Page 53: Lesson 4.1 Classifying Triangles

Lesson 4.4Proving Triangles

are CongruentToday, you will learn to…* prove that triangles are congruent* use congruence postulates to solve

problems

Page 54: Lesson 4.1 Classifying Triangles

Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent?

ASA Experiment

Page 55: Lesson 4.1 Classifying Triangles

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.

Page 56: Lesson 4.1 Classifying Triangles

A

B C

Included side?

The included side between

A and B is _____AB

Page 57: Lesson 4.1 Classifying Triangles

The included side between

B and C is _____

A

B C

CB

Included side?

Page 58: Lesson 4.1 Classifying Triangles

A

B C

ACThe included side between

A and C is _____

Included side?

Page 59: Lesson 4.1 Classifying Triangles

ASA?

1. 2.

3. 4.

ASA

ASA

NO!

NO!

Page 60: Lesson 4.1 Classifying Triangles

5. Does the diagram give enough info to use ASA Congruence?

AB C

DΔ ABD Δ by ASA

Third Angles Theorem

Reflexive Property

ACD

Page 61: Lesson 4.1 Classifying Triangles

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

Page 62: Lesson 4.1 Classifying Triangles

7. Does the diagram give enough info to use ASA Congruence?

A

B C

D

no

Reflexive Property

Page 63: Lesson 4.1 Classifying Triangles

8. Does the diagram give enough info to use ASA Congruence?

A

B

C

J

K L

KLJ _ _ _ by ASA ACB

Page 64: Lesson 4.1 Classifying Triangles

9. Determine whether the triangles are congruent by ASA.

L K

GH

J

HJG _ _ _ by ASA

Vertical Angles Theorem

Alt. Int. Angles Theorem

KJL

Page 65: Lesson 4.1 Classifying Triangles

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.

Page 66: Lesson 4.1 Classifying Triangles

AAS?

10. 11.

12. 13.

NO!

NO!

AAS

AAS

Page 67: Lesson 4.1 Classifying Triangles

14. Does the diagram give enough info to use AAS Congruence?

AB C

D

ABD _ _ _ by AAS ACDACDACD

Reflexive Property

Page 68: Lesson 4.1 Classifying Triangles

15. Does the diagram give enough info to use AAS Congruence?

A

B

C

J

K L

KLJ _ _ _ by AAS ACBACBACB

Page 69: Lesson 4.1 Classifying Triangles

16. Determine whether the triangles are congruent by AAS.

L K

GH

J

HJG _ _ _ by AAS

Vertical AnglesTheorem

Alt. Int. AnglesTheorem

K JL

Page 70: Lesson 4.1 Classifying Triangles

SSA Experiment

Using 2 sides and 1 angle that is NOT included, can you ONLY create two

triangles that are congruent?

NO

Page 71: Lesson 4.1 Classifying Triangles

AAA ExperimentUsing 3 angles, can you ONLY create

two triangles that are congruent?NO

All of the angles are , but the s are NOT

Page 72: Lesson 4.1 Classifying Triangles

Triangle Congruence?

SSS AAASSA SASASA AAS

Page 73: Lesson 4.1 Classifying Triangles

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

Page 74: Lesson 4.1 Classifying Triangles

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

Page 75: Lesson 4.1 Classifying Triangles

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

Page 76: Lesson 4.1 Classifying Triangles

What is the best way to get better at proofs?

Page 77: Lesson 4.1 Classifying Triangles
Page 78: Lesson 4.1 Classifying Triangles

Lesson 4.5Corresponding Parts of Congruent Triangles are

CongruentToday, you will learn to…* use congruence postulates to solve

problems CPCTC

Page 79: Lesson 4.1 Classifying Triangles

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

Page 80: Lesson 4.1 Classifying Triangles

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

Page 81: Lesson 4.1 Classifying Triangles

A

B

DC

3. Given: AC AD , BC BD Prove: C D

C D

Reflexive Property AB AB SSS ABC ABDCPCTC

Page 82: Lesson 4.1 Classifying Triangles

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

Page 83: Lesson 4.1 Classifying Triangles

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!

Page 84: Lesson 4.1 Classifying Triangles

Does the quilt design have vertical, horizontal, or diagonal

symmetry?

Page 85: Lesson 4.1 Classifying Triangles

Does the quilt design have vertical, horizontal, or diagonal

symmetry?

Page 86: Lesson 4.1 Classifying Triangles
Page 87: Lesson 4.1 Classifying Triangles

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.

Page 88: Lesson 4.1 Classifying Triangles

Use a ruler to draw two congruent segments that

share one endpoint.

Page 89: Lesson 4.1 Classifying Triangles

Connect the endpoints to create a triangle.

Page 90: Lesson 4.1 Classifying Triangles

Measure each interior angle. What do you notice?

base angles

legleg

base

Page 91: Lesson 4.1 Classifying Triangles

Theorem 4.6

Base Angles Theorem

If 2 sides of a triangle are congruent, then … the

angles opposite them are congruent.

Page 92: Lesson 4.1 Classifying Triangles

Theorem 4.7

Base Angles ConverseIf 2 angles of a triangle are congruent, then the sides

opposite them are congruent.

Page 93: Lesson 4.1 Classifying Triangles

B

CA D

A C by the

Base Angles Theorem

1. What angles are congruent?

Page 94: Lesson 4.1 Classifying Triangles

B

CA D

AB BC

2. What sides are congruent?

by the Base Angles

Converse

Page 95: Lesson 4.1 Classifying Triangles

3. Find mB.

A B

C

75˚

mB = 75˚

Page 96: Lesson 4.1 Classifying Triangles

4. Find mB.

A B

C

68˚

mB =68˚

44˚?

Page 97: Lesson 4.1 Classifying Triangles

5. Find x.

A B

C

2x + 4

2x + 4 = 18

18

x = 7

2x = 14

Page 98: Lesson 4.1 Classifying Triangles

6. Find x.

A

B

C

6x - 10

6x – 10 = 55

6x = 15

x = 2.5

4

Page 99: Lesson 4.1 Classifying Triangles

7. Find x and y.

50˚

x =y =

115˚ x˚65˚

6532.5

??

?

Page 100: Lesson 4.1 Classifying Triangles

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

Page 101: Lesson 4.1 Classifying Triangles

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

Page 102: Lesson 4.1 Classifying Triangles

9. Find x.

A

B

CWhat is the measure of each angle?

2x =

x = 30

2x˚

60˚

Page 103: Lesson 4.1 Classifying Triangles

10. Find x and y.

50˚

x =y =

60˚70˚

70˚

50˚

80 40

? ??

?

60˚

60˚

Page 104: Lesson 4.1 Classifying Triangles

ExperimentUsing a right angle,

a hypotenuse, and a leg, can you ONLY create 2 triangles

that are congruent?

Page 105: Lesson 4.1 Classifying Triangles

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.

Page 106: Lesson 4.1 Classifying Triangles

11. Does the diagram give enough info to use HL Congruence?

W

X

Z

Y

NO

Reflexive Property

Page 107: Lesson 4.1 Classifying Triangles

12. X is a midpoint. Does the diagram give enough info to use HL?

VWX _ _ _ by HL

V W

X

ZYYZX

Def. of midpoint

Page 108: Lesson 4.1 Classifying Triangles

13. Does the diagram give enough info to use HL Congruence? W

X

Z

Y

YWX _ _ _ by HL YZX

Base Angles Converse

Reflexive Property

Page 109: Lesson 4.1 Classifying Triangles

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