ASA- Angle Side AngleUsed to prove triangle congruence: if two angles and the included side of two triangles are congruent, then the triangles are congruent.

M

ON

LK Given: KL and NO are parallel; M bisects KO.

Prove: KLM ≡ ONM

Statements Reasons

KL and NO are parallel; M bisects KO.

KML ≡ OMN

∟MKL ≡ ∟MON

KM ≡ MO

KLM ≡ ONM

Given

Vertical Angles

Alt. Interior

Definition of bisect

ASA

Angle Angle Side -Two triangles can be proven to be congruent if two angles and the not included

side are congruent.Statements Reasons

DE=FG; DA ll EC; <B and <E are right angles

Given

<A = 90°<C = 90°

Definition right angle

<A = <C Transitive Property

EF = EF Reflexive Property

DF = GE Overlapping Segments

<D = <E Corresponding Angles Postulate

ABC = DEF AAS

• Given: DE =FG ; DA ll EC; <B and <E are right angles

• Prove: ABC = DEF

A C

B

E F GD

Side Angle Side

Statements Reasons

AB = BC; AD = EC

Given

AB = CB Segment Addition

<B = <B Reflexive Property

ABE = ___CBD

SAS

• Given: AB = BC, AD = EC• Prove: ABE = CBD

A C

F

D E

B

Hypotenuse-Leg

Statements Reasons

<1 and <2 are right angles; AB = CB

Given

<1 = 90°<2 = 90°

Definition right angle

<1 = <2 Transitive Property

BD = BD Reflexive Property

ADB = ___CDB

HL

• Given: <1 and <2 are right angles; AB = CB

• Prove: ADB = CDB

A CB

D

1 2

Side Side Side Theorem

• Given: <1= <2, <3= <4

• Prove: AFD= CFD

Statements Reasons

<1= <2, <3= <4 Given

BF=BF Reflexive

ABF= CBF ASA

AB=BC CPCTC

AF=CF CPCTC

ABC is isosceles

Def of isosceles

BD – angle bisector

Def- angle bisector

BD- perpendicular bisector

Angle bisector of the vertex angle of an isos. triangle is a perpendicular bisector of the base

AD=CD Def of perpendicular bisector

FD=FD reflexive

AFD= CFD SSS

A D C

B

F

Base Angle Theorem

• Given: AC=BC• Prove: <A=<B

Statements Reasons

AC=BC Given

DC- angle bisector

construction

<ACD=<BCD Def- angle bisector

CD=CD Reflexive

ACD= BCD

SAS

<A=<B CPCTC

A D B

C

proofs

A square is a rhombusTheorem

Statements Reasons

ABCD is a square Given

AB=BC=CD=DA Definition of a square

ABCD is a rhombus

Definition of a Rhombus

• Given: ABCD is a square Prove: ABCD is a rhombus

A B

D C

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Statements Reasons

BD bisects AC Given

BE=ED, AE=EC Definition of a bisector

<AEB=<DEC Vertical angles

AEB= CED, AED= CEB

SAS

<ECD=<EAB, <ECB=<EAD

CPCTC

AB parallel to CD,BC parallel to AD

Converse of alt. int. angles

ABCD is a parallelogram

definition

• Given: BD bisects AC• Prove: ABCD is a

parallelogram.

A B

D C

E

If one pair of adjacent sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Statements Reasons

ABCD is a parallelogram, AB=BC

Given

AB=CD, BC=AD Opposite sides of a parallelogram are congruent

CD=AB=BC=AD transitive

ABCD is a rhombus

definition

• Given: ABCD is a parallelogram, AB=BC

• Prove: ABCD is a rhombus.

A B

D C

If the diagonals of a parallelogram bisect the angles of the parallelogram, then the parallelogram is a rhombus.

Statements Reasons

ABCD is a parallelogram, BD bisects <ADC and <ABC, AC bisects <BAD and <BCD

Given

<BAE=<DAE, <ADE=<CDE, <ABE=<CBE,<BCE=<DCE

Definition of an angle bisector

<EAD=<ECB Alt. int. angles

<EAB=<ECB transitive

BE=BE reflexive

ABE= CBE AAS

AB=BC CPCTC

ABCD is a rhombus 1 pair of sides of a parallelogram are congruent.

• Given: ABCD is a parallelogram, BD bisects <ADC and <ABC, AC bisects <BAD and <BCD.

• Prove: ABCD is a rhombus.

A D

B C

E

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus:

Statements Reasons

ABCD is a parallelogram, BD perpendicular to AC.

Given

BE=BE reflexive

AE=EC Diagonals of a parallelogram bisect

<BEC=90, <BEA=90

Definition of perpendicular

<BEC=<BEA transitive

ABE= CBE SAS

AB=BC CPCTCABCD is a rhombus 1 pair of adjacent sides of

a parallelogram are congruent.

• Given: ABCD is a parallelogram, BD perpendicular to AC.

• Prove: ABCD is a rhombus.

A C

B

D

E