Triangle Theorems Notes - WPMU DEV · 2018. 2. 25. · Side-Angle-Side Similarity Postulate (SAS~)-...

SIMILARITY

THEOREMS

Similarity in Triangles

Angle-Angle Similarity Postulate (AA~)-

If two angles of one triangle are

congruent to two angles of another

triangle, then the triangles are similar.

W

R

SV

B

4545

WRS BVS

because of the AA~

Postulate.

Similarity in Triangles

Side-Angle-Side Similarity Postulate

(SAS~)- If an angle of one triangle is

congruent to an angle of a second

triangle, and the sides including the

angles are proportional, then the

triangles are similar.

TEA CUP

because of the

SAS~ Postulate.

C

U P

T

E A

32

16 1232

28 21

The scale factor is 4:3.

Similarity in Triangles

Side-Side-Side Similarity Postulate

(SSS~)- If the corresponding sides of two

triangles are proportional, then the

triangles are similar.

C

A

BQ

R S

3

4

6

1530

20

ABC QRS

because of the

SSS~ Postulate.

The scale factor is 1:5.

EXAMPLE

30°

30°

Why aren’t these triangles

congruent?

What do we call these triangles?

Ch

ris Gio

va

nello

, LB

US

D M

ath

Cu

rriculu

m O

ffice, 2

00

4

congruent polygons:

are polygons with congruent corresponding parts - their matching sides and angles

pg. 180

A X

B Y

ZC

D WPolygon ABCD Polygon XYZW

So, how do we prove

that two triangles

really are congruent?

ASA (Angle, Side, Angle)

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . .

then

the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

AAS (Angle, Angle, Side)

Special case of ASA

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . .

then

the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

SAS (Side, Angle, Side)

If in two triangles, two sides and the includedangle of one are congruent to two sides and the included angle of the other, . . .

then

the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

SSS (Side, Side, Side)

In two triangles, if 3 sides of one are congruent to three sides of the other, . . .

F

E

D

A

C

B

then

the 2 triangles are

CONGRUENT!

HL (Hypotenuse, Leg)

If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .

A

C

B

F

E

D

then

the 2 triangles are

CONGRUENT!

HA (Hypotenuse, Angle)

If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .

then

the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

LA (Leg, Angle)

If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .

then

the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

LL (Leg, Leg)

If both pair of legs of two RIGHT triangles are congruent, . . .

then

the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

Example 1

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

F

E

D

A

C

B

Example 2

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

A

C

B

F

E

D

Ch

ris Gio

va

nello

, LB

US

D M

ath

Cu

rriculu

m O

ffice, 2

00

4

CPCTC:

corresponding parts of congruent triangles are

congruent

pg. 203

The angle between two sides

Included Angle

HGI G

GIH I

GHI H

This combo is called side-angle-side, or just SAS.

19

Name the included angle:

YE and ES

ES and YS

YS and YE

Included Angle

SY

E

YES or E

YSE or S

EYS or Y

The other two angles are the

NON-INCLUDED angles. 20

The side between two angles

Included Side

GI HI GH

This combo is called angle-side-angle, or just ASA. 21

Name the included side:

Y and E

E and S

S and Y

Included Side

SY

E

YE

ES

SY

The other two sides are the

NON-INCLUDED sides. 22

TRIANGLE PROPORTIONALITY

THEOREM

23

CONVERSE OF TRIANGLE

PROPORTIONALITY THEOREM

24

TRIANGLE MIDSEGMENT THEOREM

25

KEY CONCEPTS

A transversal is a line that cuts through two parallel

lines.

When a triangle contains a line that is parallel to one

of its sides, the two triangles formed can be proved

similar using the AA Similarity Postulate. Since

triangles are similar, their sides are proportional.

The three midsegments of a triangle form the

Midsegment Triangle.

26

Are the following triangles similar?

If so, what similarity statement can

be made. Name the postulate or

theorem you used.

F

G

H

K

J

Yes, FGH KJH because

of the AA~ Postulate

Are the following triangles similar?

If so, what similarity statement can

be made. Name the postulate or

theorem you used.M

O R

G

H I6

10

3

4

No, these are not similar

because

Are the following triangles similar?

If so, what similarity statement can

be made. Name the postulate or

theorem you used.

A

X Y

B C

20

25

25

30

No, these are not similar

because

Are the following triangles similar?

If so, what similarity statement can

be made. Name the postulate or

theorem you used.

Yes, APJ ABC because of

the SSS~ Postulate.

A

P J

B C

3

5

2

3

8

3

Explain why these triangles are

similar. Then find the value of x.

3

5

4.5

x

These 2 triangles are similar

because of the AA~ Postulate.

x=7.5

Explain why these triangles are

similar. Then find the value of x.

These 2 triangles are similar

because of the AA~ Postulate.

x=2.5

5

70 1103 3

x

Explain why these triangles are

similar. Then find the value of x.

22

1424

x

These 2 triangles are similar

because of the AA~ Postulate.

x=12

Explain why these triangles are

similar. Then find the value of x.

These 2 triangles are similar

because of the AA~ Postulate.

x= 12

x

6

2

9

Explain why these triangles are

similar. Then find the value of x.

These 2 triangles are similar

because of the AA~ Postulate.

x=8

15

4

x

5

Explain why these triangles are

similar. Then find the value of x.

These 2 triangles are similar

because of the AA~ Postulate.

x= 15

18

7.5 12

x

Please complete the Ways to

Prove Triangles Similar

Worksheet.

Side Splitter Theorem - If a line is parallel

to one side of a triangle and intersects the

other two sides, then it divides those sides

proportionally.

Similarity in Triangles

T

S U

R V

x 5

1610

You can either

use

or

Theorem

If three parallel lines intersect two

transversals, then the segments

intercepted are proportional.

a

b

c

d

Theorem

Triangle Angle Bisector Theorem -If a

ray bisects an angle of a triangle, then it

divides the opposite side on the triangle

into two segments that are proportional to

the other two sides of the triangle.

A

BC D