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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 1. 2. 3. 4. 5. 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 Shared side SSS Parallel lines -> AIA Vertical angles SAS Shared side SAS 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) 1. AB DE 2. BC EF 3. AC DF ABC DEF Side-Angle-Side (SAS) 1. AB DE 2. A D 3. AC DF ABC DEF included angle Angle-Side-Angle (ASA) 1. A D 2. AB DE ABC DEF 3. B E include d side Angle-Angle-Side (AAS) 1. A D 2. B E ABC DEF 3. BC EF Non-included side Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C D NOT CONGRUENT F Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT Name That Postulate (when possible) SAS SSA ASA 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. HL ASA Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA Name That Postulate (when possible) Name That Postulate (when possible) Closure Question Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F 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 K I H J 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. Ex 5 B A C D E 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 M K L 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 K L V Not possible U 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 AB Side PQ BC QR Side AC PR Then ∆ABC ≅ ∆PQR Example 1 Prove: ∆DEF ≅ ∆JKL From the diagram, DE JK , DF JL, and EF KL. ∆DEF ≅ ∆JKL SSS Congruence Postulate. 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 AC DF 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 A areQ congruent. R S B C 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 A Q congruent. R S B C AAS Proof • If 2 angles are congruent, so is the 3rd • Third Angle Theorem A • NowQ QR is an included side, so ASA. R S B C 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? 1 2 4 3 Example • Is it possible to prove these triangles are congruent? 1 2 4 3 • 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. Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y Included Side The side between two angles GI HI GH Included Side Name the included side: E Y S Y and E YE E and S ES S and Y 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 MN QR, NP RS , PM SQ then, MNP QRS Using SSS Congruence Post. Prove: PQW TSW • 1) PQ ST , QW WS , PW TW • 2) PQW TSW • 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 PQ WX , QS XY , Q X then, PQS WXY Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y Included Side The side between two angles GI HI GH Included Side Name the included side: E Y S Y and E YE E and S ES S and Y SY Triangle congruency shortC cuts B A D F E H 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 G SAS J I Given: 1 2, A E and AC EC B Prove: ΔABC ΔEDC 1 2 A E AC EC Given Given Given ΔABC ΔEDC D 1 A ASA 2 C E Given: ΔABD, ΔCBD, AB CB, and AD CD Prove: ΔABD ΔCBD A AB CB AD CD BD BD C Given Given Reflexive Prop ΔABD ΔCBD B SSS D Given: LJ bisects IJK, ILJ JLK Prove: ΔILJ ΔKLJ I J L LJ bisects IJK IJL IJH ILJ JLK JL JL Given K Definition of bisector Given Reflexive Prop ΔILJ ΔKLJ ASA Given: TV VW, UV VX Prove: ΔTUV ΔWXV TV VW UV VX TVU WVX U Given T Given Vertical angles W V X ΔTUV ΔWXV SAS Given: Given: HJ JL, H L Prove: ΔHIJ ΔLKJ I HJ JL H L IJH KJL K Given Given Vertical angles J H ΔHIJ ΔLKJ ASA L Given: Quadrilateral PRST with PR ST, PRT STR Prove: ΔPRT ΔSTR PR ST PRT STR RT RT R S Given Given Reflexive Prop P ΔPRT ΔSTR SAS T Given: Quadrilateral PQRS, PQ QR, PS SR, and QR SR Prove: ΔPQR ΔPSR PQ QR PQR = 90° PS SR PSR = 90° QR SR PR PR Given P PQ QR Given PS SR Given Reflexive Prop ΔPQR ΔPSR HL Q R S Prove it! NOT triangle congruency short cuts NOT triangle congruency short-cuts A • • • The following are NOT short cuts: AAA (angle-angleangle) Triangles are similar but not necessarily congruent 60 60 C 60 D B 60 60 F 60 E NOT triangle congruency B short-cuts • • • The following are NOT short cuts SSA (side-sideangle) SAS is a short cut but the angle is in between both sides! 8 cm 5 cm 34 A E 8 cm 5 cm 34 F D C 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 U Given: ΔTUV, ΔWXV, TV WV, TW bisects UX T V Prove: TU WX Statements: Reasons: X 1. TV WV Given 2. UV VX Definition of bisector 3. TVU WVX Vertical angles are congruent 4. ΔTUV ΔWXV SAS 5. TU WX CPCTC W Side Side Side If 2 triangles have 3 corresponding pairs of sides that are congruent, then the X ~ AC = PX triangles are congruent. AB ~ = PN ~ CB = XN A P C N B Therefore, using SSS, ~ ∆ABC = ∆PNX 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 60° A C 60° ~ CA = XP ~ CB = XN ~ C = X P N B Therefore, by SAS, ~ = ∆ABC ∆PNX 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 ~ A 70 ° C 60° CA = XP ~ A = P ~ C = X 60° 70 ° N B P Therefore, by ASA, ~ ∆ABC= ∆PNX 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° 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 . B F That means that EG CB A C E G What is AC congruent to? FE Corresponding parts of congruent triangles are congruent. Corresponding parts of congruent triangles are congruent. Corresponding parts of congruent triangles are congruent. 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: A Prove: AB DE Statements B C D AC DF Given C F Given CB FE Given ΔABC ΔDEF AB DE F E Reasons SAS CPCTC Using SAS Congruence Prove: Δ VWZ ≅ Δ XWY PROOF WZ WY ,VW XW Given VWZ XWY Vertical Angles Δ VWZ ≅ Δ XWY SAS Proof Given: MB is perpendicular bisector of AP Prove: • • • • • • ABM PBM 1) MB is perpendicular bisector of AP 2) <ABM and <PBM are right <‘s 3) AB BP 4) ABM PBM 5) BM BM 6) ABM PBM • • • • • • 1) Given 2) Def of Perpendiculars 3) Def of Bisector 4) Def of Right <‘s 5) Reflexive Property 6) SAS Proof Given: O is the midpoint of MQ and NP Prove: • • • • MON POQ 1) O is the midpoint of MQ and NP 2) MO OQ, NO OP 3) MON POQ 4) MON POQ • • • • 1) Given 2) Def of midpoint 3) Vertical Angles 4) SAS Proof Given: Prove: AB CD, BC AD ABC ADC • 1) AB CD, BC AD • 2) AC AC • 3) ABC ADC • 1) Given • 2) Reflexive Property • 3) SSS Proof Given: Prove: • • • • AD CB, AD || CB ABD CDB 1) AD CB, AD || CB 2) ADB CBD 3) DB DB 4) ABD CDB • • • • 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? EXAMPLE 2 Standardized Test Practice SOLUTION By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR. d = ( x2 – x1 ) 2 + ( y2 – y1 ) 2 Write a proof. GIVEN PROVE Proof KL NL, KM KLM NM NLM It is given that KL NL and KM By the Reflexive Property, LM So, by the SSS Congruence Postulate, KLM NLM NM LM. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 1. 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, Yes. The statement is true. DFG HJK. Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y In the diagram at the right, what postulate or theorem can you use to prove that RST S RS RTS VUT Given U Given UV UTV Vertical angles Δ RST ≅ Δ VUT SAA Now For The Fun Part… Given: JO SH; O is the midpoint of SH Prove: SOJ HOJ J S 0 H Given: BC bisects AD A D Prove: AB DC A C E B D