5-5 & 5-6 SSS, SAS, ASA, & AAS

Proving Triangles Congruent

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

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

Corresponding Parts You learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C AB  DE BC  EF AC  DF  A   D  B   E  C   F ABC   DEF E D F

Do you need all six ? NO ! SSS SAS ASA AAS

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

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

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

Included Angle Name the included angle: YE and ES ES and YS YS and YE

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

Included Side The side between two angles GI GH HI

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

Angle-Angle-Side (AAS) B E F A C D A   D  B   E BC  EF ABC   DEF Non-included side

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

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

The Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence

Name That Postulate (when possible) SAS ASA SSA SSS

Name That Postulate (when possible) AAA ASA SSA SAS

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

HW: Name That Postulate (when possible)

HW: Name That Postulate (when possible)

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

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

Write a congruence statement for each pair of triangles represented.   B D A C E F  

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5.6 ASA and AAS

Before we start…let’s get a few things straight C X Z Y INCLUDED SIDE

Angle-Side-Angle (ASA) Congruence Postulate Two angles and the INCLUDED side

Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included

Your Only Ways To Prove Triangles Are Congruent SSS SAS ASA AAS Your Only Ways To Prove Triangles Are Congruent

Alt Int Angles are congruent given parallel lines Things you can mark on a triangle when they aren’t marked. Overlapping sides are congruent in each triangle by the REFLEXIVE property Alt Int Angles are congruent given parallel lines Vertical Angles are congruent

Ex 1 DEF NLM

Ex 2 What other pair of angles needs to be marked so that the two triangles are congruent by AAS? F D E M L N

Ex 3 What other pair of angles needs to be marked so that the two triangles are congruent by ASA? F D E M L N

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 G I H J K ΔGIH  ΔJIK by AAS

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. B A C E D Ex 5 ΔABC  ΔEDC by ASA

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB  ΔECD by SAS

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 7 J K L M ΔJMK  ΔLKM by SAS or ASA

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 8 J T L K V U Not possible

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(over Lesson 5-5) Slide 1 of 2 1-1a

(over Lesson 5-5) Slide 1 of 2 1-1b

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