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