SUMMARY OF REASONS: GRADE 101. EQUAL ANGLES

1 3

2 4

and are straight lines intersecting at E

ˆ ˆvert. opp s

ˆ ˆ

AB CD

E E

E E

43

21E

D

C

A

In

ˆ ˆ opp equal sides

ABC AB AC

C B s ABC

C

B

A

ˆˆ ˆ ˆIn s and and

ˆ ˆ 3rd s of s

ABC FED B E C D

A F

A

C

BD E

al ||| (corr. sides prop )

ˆ ˆ

ˆ ˆ corr. s similar s

ˆ ˆ

ABC FED

A F

B E

C D

p

nm

kp

kn

km

A

C

BD E

(SSS or SAS or RHS)

ˆ ˆ

(or any other pairs of corr. s not given equal)

ABC DEF

A D

F

E

DC

B

A

cuts at and EH AB CD F G

1 1

2 2

3 3

4 4

ˆˆ

ˆˆ(corr. s )

ˆˆ

ˆˆ

F G

F GAB CD

F G

F G

2

4

43

32

1

1

H

G

F

E

DC

BA

cuts at and EH AB CD F G

2 4

4 2

ˆˆ(alt. s )

ˆˆ

F GAB CD

F G

2

4

43

32

1

1

H

G

F

E

DC

BA

is a parallelogramPQRS

ˆ ˆ

opp. s gram ˆ ˆ

P RPQRS

Q S

S

RQ

P

1 2

1 2

1 2

1 2

is a rhombus with diagonals and

ˆ ˆ

ˆ ˆdiags. of rhombus bisect s

ˆ ˆ

ˆ ˆ

RHOM RO HM

R R

H H

O O

M M

M

OH

2

22

21

11

1R

is a kite with and KITE KI KE TI TE

1 2

1 2

ˆ ˆone diag. of kite bisects s

ˆ ˆ

K K

T T

E

T

K

I

2

2

1

1

2. EQUAL LINE SEGMENTS

(sides opp. equal angles in ABC)CA CB A

C

B

If (RHS or SAS or AAS)ABC FED (corr sides ABC FED)AB FE This applies to all corresponding

sides not used in proving congruence

A

C

B

F

D E

is a parallelogram

opp. sides || gram

PQRS

PQ SRPQRS

QR PS

S

RQ

P

is a parallelogram

Diagonals intersect at T

diags of || gram

PQRS

PT TRPQRS

QT TS

T

S

RQ

P

is a rectangle

(diags. rectangle RECT)

RECT

RC ET

T

CE

R

is a kite with and KITE KI KE

TI TE

(bisected diag. of kite KITE)EM MII

M

E

T

K

In

and ||

ABC

AD DB DE BC

AE EC

ED

CB

A

3. LINES PARALLEL

1 1

2 2

3 3

4 4

ˆˆ

ˆˆor Given or proved

ˆˆor

ˆˆor

F G

F G

F G

F G

But these are corr. s

||AB CD

2

4

43

32

1

1

H

G

F

E

DC

BA

2 4

3 1

ˆˆ Given or proved

ˆˆor

But these are alt. s

||

F G

F G

AB CD

2

4

43

32

1

1

H

G

F

E

DC

BA

02 1

03 4

ˆˆ 180Given or proved

ˆˆor 180

But these are co-int. s

||

F G

F G

AB CD

2

4

43

32

1

1

H

G

F

E

DC

BA

In

(Given or proved)

(Given or proved)

||

ABC

AD DB

AE EC

DE BC

ED

CB

A

is a || gram / rhom/rect/squareABCD

||Opp sides || gram/rhom/rect/square

||

AD BC

AB DC

D

C

A

B

|| (Given or proved)

|| (Given or proved)

|| (Both || )

AB CD

EF CD

AB EF CD

F

E

D

C

B

A

4. PARALLELOGRAMS

|| (Given or proved)

|| (Given or proved)

is a || gram (Both prs opp sides ||)

AD BC

AB DC

ABCD

D

C

A

B

(Given or proved)

(Given or proved)

is a || gram (Both prs opp sides equal)

AD BC

AB DC

ABCD

D

C

A

B

ˆ ˆ (Given or proved)

ˆ ˆ (Given or proved)

is a || gram (Both prs opp angles equal)

A C

B D

ABCD

D

C

A

B

(Given or proved)

(Given or proved)

is a || gram (Diags. bisect each other)

AE EC

BE ED

ABCD

E

D

C

A

B

(Given or proved)

and || (Given or proved)

OR

(Given or proved)

and || (Given or proved)

is a || gram (One pr. opp sides equal and ||)

AD BC

AD BC

AB DC

AB DC

ABCD

D

C

A

B

5. OTHER DEDUCTIONS

01 2

is a straight line

ˆ ˆ 180 adj. s on str. line

ABC

B B ABC 2

1

D

C

B

A

02

ˆ ˆ 180 given or proved

But these are adj. s

is a straight line

B B

ABC

2

1

D

C

B

A

2 1

03 4

|| is cut by transversal

ˆˆ 180co-int. s ||

ˆˆ 180

AB CD EFGH

F GAB CD

F G

2

4

43

32

1

1

H

G

F

E

DC

BA

0ˆ ˆˆ 180 3 of A B C s ABC

C

B

A

1

Side of is produced to

ˆ ˆˆ (Ext of = sum int. opp. s)

CA ABC D

A B C

21D

C

B

A

01

If is a rhombus/square/kite

ˆ diags of rhombus/square/kite 90

cut at rt s or

ABCD

E

AC BD

1

E

D

C

A

B

In and

12 or

2

ABC AD DB AE EC

BC DE DE BC

ED

CB

A

6. CONGRUENCE

In s and

Given or proved

SSS

ˆ ˆ

ˆ ˆ Corr. parts

ˆ ˆ

ABC DEF

AB DE

BC EF

AC DF

ABC DEF

A D

B E ABC DEF

C F

E

D

B

A

C

F

In s and

ˆ ˆ Given or proved

SAS

ˆ ˆ Corr. parts

ˆ ˆ

ABC DEF

AB DE

A F

AC DF

ABC DEF

BC EF

B E ABC DEF

C F

D

B

AE

C

F

0

In s and

ˆ ˆ 90

Given or proved

or

RHS

or Corr. parts ˆ ˆ

ˆ ˆ

ABC DFE

B F

AC DE

AB DF BC EF

ABC DEF

BC EF

AB DFABC DEF

C E

A D

B

AE

DC

F

In s and

or

or Given or provedˆ ˆ

ˆ ˆand

AAS

and/or

Corr. parts

and/or

ABC DEF

AB DE

AC DF

BC EF

A D

B E

ABC DEF

AB DE

BC EF ABC DEF

AC DF

B

AE

DC

F

7. SIMILARITY

In s and

if two angles are equal

|||

and so

1. Third angles are equal

2. Corresponding sides are in proportion

ABC DEF

any

ABC DEF

A

C

B

F

D E

2

In s and

if are in proportion:

: : : :

|||

and so

1. The corr. angles are equal

2. If then

area area

ABC DEF

all sides

AB BC AC DE EF DF

ABC DEF

DE kAB

DEF k DEF

p

nm

kp

kn

km

A

C

B

F

D E

In all polygons except s,

for polygons to be similar:

1. angles must be equal

2. sides must be in proportion

all and

all

So rect. is similar to

rect. unless

: :

ABCD not

EFGH

AB BC EF FG

H

GF

E

D

CB

A