Proving Triangles Congruent

Two geometric figures with exactly the same size and shape. The Idea of a Congruence A C B DE F

How much do you need to know... need to know about two triangles to prove that they are congruent?

If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Corresponding Parts  ABC   DEF B A C E D F 1.AB  DE 2.BC  EF 3.AC  DF 4.  A   D 5.  B   E 6.  C   F

Do you need all six ? NO ! SSS SAS ASA RHS S means side & A means angle

1) Side-Side-Side (SSS) 1. AB  DE 2. BC  EF 3. AC  DF  ABC   DEF B A C E D F

2) Side-Angle-Side (SAS) 1. AB  DE 2.  A   D 3. AC  DF  ABC   DEF B A C E D F included angle

The angle between two sides Included Angle  G G  I I  H H

Name the included angle: YE and ES ES and YS YS and YE Included Angle SY E  E E  S S  Y Y

3) Angle-Side- Angle (ASA) 1.  A   D 2. AB  DE 3.  B   E  ABC   DEF B A C E D F included side

The side between two angles Included Side GI HI GH

Name the included side :  Y and  E  E and  S  S and  Y Included Side SY E YE ES SY

Angle-Angle-Side (AAS) 1.  A   D 2.  B   E 3. AC  EF NOT  B A C E D F Non-included side

Angle-Angle-Side (AAS) Sometimes will be ( ASA ) 1.  A   D 2.  B   E 3. BC  EF  ABC   DEF B A C E D F  C   F ( why ?) * *

4) Right angle- hypotenuse - side (RHS) 4) Right angle - hypotenuse - side (RHS) 1.  A =  D = AB  DE 3. BC  EF  ABC   DEF B A C E D F ∘

Warning: No SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!

Warning: No AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT

The Congruence Postulates  SSS correspondence  ASA correspondence  SAS correspondence  RHS correspondence  SSA correspondence  AAA correspondence

Example In the given figure prove that : AB ≅ AD B C D A In ∆∆ ADC, ABC we have : فيهما : ADC, ABC ∆∆ 1) DC = BC ( given ) 1) ( معطى ) DC = BC 2) ∡ DCA = ∡ BCA ( given ) 2) ( معطى ) ∡ DCA = ∡ BCA 3) AC = AC ( common side ) 3) ( ضلع مشترك )AC = AC ∴ ∆ ACD ≅ ∆ ACB ( SAS postulate) ∴ ∆ ACD ≅ ∆ ACB ( SAS مسلمة ) ∴ AB ≅ AD ( CPCTC ) Corresponding parts of ≅ ∆ s are ≅ ∴ AB ≅ AD ( ( الأجزاء المتناظرة في المثلثات المتطابقة تكون متطابقة

Name That Postulate SAS ASA SSS SSA (when possible)

Name That Postulate (when possible) ASA SAS AAA SSA

Name That Postulate (when possible) SAS SAS SAS Reflexive Property Vertical Angles Reflexive Property SSA

ACTIVITY : Name That Postulate (when possible)

ACTIVITY : Name That Postulate

Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS:  B   D For AAS ASA : A  F A  F AC  FE

ACTIVITY Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS ASA :