Lesson 4.4 - 4.5 Proving Triangles Congruent

Triangle Congruency Short-Cuts If you can prove one of the following short cuts, you have two congruent triangles SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-angle-side) HL (hypotenuse-leg) right triangles only!

Built – In Information in Triangles  

Identify the ‘built-in’ part

SAS SAS SSS Shared side Vertical angles Parallel lines -> AIA

SOME REASONS For Indirect Information Def of midpoint Def of a bisector Vert angles are congruent Def of perpendicular bisector Reflexive property (shared side) Parallel lines ….. alt int angles Property of Perpendicular Lines

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

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

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

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

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

Name That Postulate (when possible) SAS ASA SSA SSS

This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. ASA HL

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

Name That Postulate (when possible)

Name That Postulate (when possible)

Closure Question

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:

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. J T Ex 8 L K V U Not possible

SSS (Side-Side-Side) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent. If Side Side Side Then ∆ABC ≅ ∆PQR

Example 1 SSS Congruence Postulate. Prove: ∆DEF ≅ ∆JKL From the diagram, SSS Congruence Postulate. ∆DEF ≅ ∆JKL

SAS (Side-Angle-Side) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, the two triangles are congruent. If Angle ∠A ≅ ∠D Side Angle ∠C ≅ ∠F Then ∆ABC ≅ ∆DEF

Example 2 Prove: ∆SYT ≅ ∆WYX

Side-Side-Side Postulate SSS postulate: If two triangles have three congruent sides, the triangles are congruent.

Angle-Angle-Side Postulate If two angles and a non included side are congruent to the two angles and a non included side of another triangle then the two triangles are congruent.

Angle-Side-Angle Postulate If two angles and the side between them are congruent to the other triangle then the two angles are congruent.

Side-Angle-Side Postulate If two sides and the adjacent angle between them are congruent to the other triangle then those triangles are congruent.

Which Congruence Postulate to Use? 1. Decide whether enough information is given in the diagram to prove that triangle PQR is congruent to triangle PQS. If so give a two-column proof and state the congruence postulate.

ASA If 2 angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the 2 triangles are congruent. A Q S C R B

AAS If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent. A Q S C R B

AAS Proof If 2 angles are congruent, so is the 3rd Third Angle Theorem Now QR is an included side, so ASA. A Q S C R B

Example Is it possible to prove these triangles are congruent?

Example Is it possible to prove these triangles are congruent? Yes - vertical angles are congruent, so you have ASA

Example Is it possible to prove these triangles are congruent?

Example Is it possible to prove these triangles are congruent? No. You can prove an additional side is congruent, but that only gives you SS

Example Is it possible to prove these triangles are congruent? 2 1 3 4

Example Is it possible to prove these triangles are congruent? Yes. The 2 pairs of parallel sides can be used to show Angle 1 =~ Angle 3 and Angle 2 =~ Angle 4. Because the included side is congruent to itself, you have ASA. 2 1 3 4

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

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

Side-Side-Side Congruence Postulate SSS Post. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If then,

Using SSS Congruence Post. Prove: 1) 2) 1) Given 2) SSS

Side-Angle-Side Congruence Postulate SAS Post. – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If then,

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

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

Triangle congruency short-cuts  

Given: HJ  GI, GJ  JI Prove: ΔGHJ  ΔIHJ HJ  GI Given GJH & IJH are Rt <‘s Def. ┴ lines GJH  IJH Rt <‘s are ≅ GJ  JI Given HJ  HJ Reflexive Prop ΔGHJ  ΔIHJ SAS

ΔABC  ΔEDC ASA Given: 1  2, A  E and AC  EC Prove: ΔABC  ΔEDC 1  2 Given A  E Given AC  EC Given ΔABC  ΔEDC ASA

Given: ΔABD, ΔCBD, AB  CB, and AD  CD Prove: ΔABD  ΔCBD AB  CB Given AD  CD Given BD  BD Reflexive Prop  ΔABD  ΔCBD SSS

Given: LJ bisects IJK, ILJ   JLK Prove: ΔILJ  ΔKLJ LJ bisects IJK Given IJL  IJH Definition of bisector ILJ   JLK Given JL  JL Reflexive Prop ΔILJ  ΔKLJ ASA

 ΔTUV  ΔWXV SAS Given: TV  VW, UV VX Prove: ΔTUV  ΔWXV TV  VW Given UV  VX Given TVU  WVX Vertical angles  ΔTUV  ΔWXV SAS

 ΔHIJ  ΔLKJ ASA Given: Given: HJ  JL, H L Prove: ΔHIJ  ΔLKJ HJ  JL Given H L Given IJH  KJL Vertical angles  ΔHIJ  ΔLKJ ASA

ΔPRT  ΔSTR SAS Given: Quadrilateral PRST with PR  ST, PRT  STR Prove: ΔPRT  ΔSTR PR  ST Given PRT  STR Given RT  RT Reflexive Prop ΔPRT  ΔSTR SAS

ΔPQR  ΔPSR HL Given: Quadrilateral PQRS, PQ  QR, PS  SR, and QR  SR Prove: ΔPQR  ΔPSR PQ  QR Given PQR = 90° PQ  QR PS  SR Given PSR = 90° PS  SR QR  SR Given PR  PR Reflexive Prop ΔPQR  ΔPSR HL

NOT triangle congruency short cuts Prove it! NOT triangle congruency short cuts

NOT triangle congruency short-cuts The following are NOT short cuts: AAA (angle-angle-angle) Triangles are similar but not necessarily congruent

NOT triangle congruency short-cuts The following are NOT short cuts SSA (side-side-angle) SAS is a short cut but the angle is in between both sides!

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Prove it! CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

CPCTC Once you have proved two triangles congruent using one of the short cuts, the rest of the parts of the triangle you haven’t proved directly are also congruent! We say: Corresponding Parts of Congruent Triangles are Congruent or CPCTC for short

CPCTC example Given: ΔTUV, ΔWXV, TV  WV, TW bisects UX Prove: TU  WX Statements: Reasons: TV  WV Given UV  VX Definition of bisector TVU  WVX Vertical angles are congruent ΔTUV  ΔWXV SAS TU  WX CPCTC

Side Side Side If 2 triangles have 3 corresponding pairs of sides that are congruent, then the triangles are congruent. X P N 3 inches 7 inches 5 inches AC PX AB PN CB XN Therefore, using SSS, ∆ABC = ∆PNX ~ = A B C 3 inches 7 inches 5 inches

Side Angle Side If two sides and the INCLUDED ANGLE in one triangle are congruent to two sides and INCLUDED ANGLE in another triangle, then the triangles are congruent. X P N 3 inches 5 inches 60° CA XP CB XN C  X Therefore, by SAS, ∆ABC ∆PNX = ~ A B C 3 inches 5 inches 60°

Angle Side Angle If two angles and the INCLUDED SIDE of one triangle are congruent to two angles and the INCLUDED SIDE of another triangle, the two triangles are congruent. X CA XP A P C X Therefore, by ASA, ∆ABC ∆PNX = ~ 60° A 3 inches 3 inches 70° 70° P N 60° C B

Side Angle Angle Triangle congruence can be proved if two angles and a NON-included side of one triangle are congruent to the corresponding angles and NON-included side of another triangle, then the triangles are congruent. 60° 70° 60° 70° 5 m 5 m These two triangles are congruent by SAA

Remembering our shortcuts SSS ASA SAS SAA

Corresponding parts When you use a shortcut (SSS, AAS, SAS, ASA, HL) to show that 2 triangles are , that means that ALL the corresponding parts are congruent. EX: If a triangle is congruent by ASA (for instance), then all the other corresponding parts are . A C B G E F That means that EG  CB FE What is AC congruent to?

Corresponding parts of congruent triangles are congruent.

Corresponding Parts of Congruent Triangles are Congruent. CPCTC If you can prove congruence using a shortcut, then you KNOW that the remaining corresponding parts are congruent. You can only use CPCTC in a proof AFTER you have proved congruence.

For example: Prove: AB  DE A Statements Reasons B C AC  DF Given C  F Given CB  FE Given ΔABC  ΔDEF SAS AB  DE CPCTC D F E

Using SAS Congruence Prove: Δ VWZ ≅ Δ XWY Δ VWZ ≅ Δ XWY SAS PROOF Given Δ VWZ ≅ Δ XWY Vertical Angles SAS

Proof 1) MB is perpendicular bisector of AP Given: MB is perpendicular bisector of AP Prove: 1) MB is perpendicular bisector of AP 2) <ABM and <PBM are right <‘s 3) 4) 5) 6) 1) Given 2) Def of Perpendiculars 3) Def of Bisector 4) Def of Right <‘s 5) Reflexive Property 6) SAS

Proof 1) O is the midpoint of MQ and NP 2) 3) 4) 1) Given Given: O is the midpoint of MQ and NP Prove: 1) O is the midpoint of MQ and NP 2) 3) 4) 1) Given 2) Def of midpoint 3) Vertical Angles 4) SAS

Proof Given: Prove: 1) 2) 3) 1) Given 2) Reflexive Property 3) SSS

Proof 1) 2) 3) 4) 1) Given 2) Alt. Int. <‘s Thm Prove: 1) 2) 3) 4) 1) Given 2) Alt. Int. <‘s Thm 3) Reflexive Property 4) SAS

Checkpoint Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use.

Congruent Triangles in the Coordinate Plane Use the SSS Congruence Postulate to show that ∆ABC ≅ ∆DEF Which other postulate could you use to prove the triangles are congruent?

Congruent Triangles

Standardized Test Practice EXAMPLE 2 Standardized Test Practice SOLUTION By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR. d = y 2 – 1 ( ) x +

Write a proof. GIVEN KL NL, KM NM PROVE KLM NLM Proof It is given that KL NL and KM NM By the Reflexive Property, LM LM. So, by the SSS Congruence Postulate, KLM NLM

GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. DFG HJK SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Side DG HK, Side DF JH,and Side FG JK. So by the SSS Congruence postulate, DFG HJK. Yes. The statement is true.

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

In the diagram at the right, what postulate or theorem can you use to prove that RST VUT Given S U Given RS UV Δ RST ≅ Δ VUT SAA Vertical angles RTS UTV

Now For The Fun Part… Proofs!

Given: JO  SH; O is the midpoint of SH Prove:  SOJ  HOJ

Given: BC bisects AD A   D Prove: AB  DC A C E B D

WORK