Triangle Congruencies

HOMEWORK: Proofs WS & Triangle Packet - evensQuiz Friday

January11, 2012Lessons 4.3 – 4.5

Triangle Sum, Triangle Inequalities, Side-Angle, Triangle Congruency

Two Triangles are congruent if they are the same shape and size.

To PROVE two triangles congruent means to show that all corresponding parts are congruent.

Each triangle has 6 parts: 3 sides & 3 angles

Congruent Triangles

However, shortcuts have been discovered and they make our job easier.

SSSAAASASSSAASASAA

Possible Triangle Congruency Conjecture

Combinations

S – SIDE

A – ANGLE

ORDER MATTERS!!

The order of the corresponding parts in the triangles must mimic the order of the letters in the Congruency Statements

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Three pairs of congruent sides

SSSAAASASSSAASASAA

Possible Triangle

Congruency Combinations

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Three pairs of congruent sides

YES!

SSS

√SSSAAASASSSAASASAA

90°

65°25°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Three pairs of congruent angles

SSSAAASASSSAASASAA

√

90°

65°

25° 25°90°

65°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Three pairs of congruent angles

NO!

All Equilateral/Equiangular triangles have three 60° angles.

60°

60°60°

60°

60°

60°

60°

60°60°

A Counterexample to an AAA congruence conjecture

SSSAAASASSSAASASAA

√

Notes - Congruent TrianglesAre these guarantees of triangle

congruence?

So far this is what has been

determined from the list

Now let’s try combinations with 2 sides and an angle

25°

SSSAAASASSSAASASAA

√

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?One pair of congruent angles, two pairs of

congruent sides

25°

25°√

SSSAAASASSSAASASAA

√

25°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and the included angle

YES!SAS!

An included angle is between the two given sides.

A

BC

ABC DEF

92°

D

E

F

92°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and the included angle

√

SSSAAASASSSAASASAA

√

65°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and a non-included

angle

√

SSSAAASASSSAASASAA

√

65°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and a non-included

angle

√

SSSAAASASSSAASASAA

√

65°

√

SSSAAASASSSAASASAA

√

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and a non-included

angle

65°

√

SSSAAASASSSAASASAA

√

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and a non-included

angle

A

BC

D

E

F

ABC DEF

√

SSSAAASASSSAASASAA

√

NO!

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent sides and a non-included

angle

90°

25°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent angles and the included side

√

SSSAAASASSSAASASAA

√

90°25°

90°25°

√

SSSAAASASSSAASASAA

√

YES! ASA!

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent angles and the included side

65°55°

A

B

C

65°55°

D

E

F

ABC DEF

ASA!

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent angles and the included side

√

SSSAAASASSSAASASAA

√

√

90° 35°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent angles and the non-included

side

√

SSSAAASASSSAASASAA

√

√

35°

90°

35°

35°

Notes - Congruent TrianglesIs this a guarantee of triangle

congruence?Two pairs of congruent angles and the non-included

side

√

SSSAAASASSSAASASAA

√

√√ NO

NO

YES!

√

SSSAAASASSSAASASAA

√

√√

Notes - Congruent TrianglesThese are the triangle congruencies.

SSS

SAS

ASASAA

Using Properties of Right triangles In Combination with Triangle Congruencies

Hypotenuse –Leg Congruence Theorem (HL)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

One More Congruency

HL (Hypotenuse - Leg)is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate. 

HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!

SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

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

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

HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

Methods of Proving Triangles Congruent

Congruent TrianglesName the congruence

ASA

S

F

R

120°

35°D

P

Q120°

35°

FRS PQD

Is ?

SSA

FRS QSR ?

S

F

R

42°

42°Q

Congruent TrianglesName the congruence

Shared Side – Reflexive Prop

Is

WHY NOT?SSA is NOT a valid Triangle congruence

SAS

FRS QSR

S

F

R

50°

50°Q

Congruent TrianglesName the congruence

Shared Side – Reflexive Prop

Is ?

SSS ?

MNR PTB

M

N

R

T

P

B

Names of the triangles in the congruence statement are not in corresponding order.

WHY NOT?

Congruent TrianglesName the congruence

Is ?