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Triangles Congruent Triangles Define • Congruent Polygons: • Congruent Triangles: • Reflection: • Reflexive Property: • Rotate: • Slide: Naming Congruent Parts 1. If IiHIJ L,.PQR, name the congruent angles and sides. Q if I 2. If tDEF AABC, name the congruent angles and sides. R D Identifying Congruent Triangles Complete the congruency statements. 3. A 4. a D H Z L ? D L\GIHL &‘DE Transformations Movement Flip Slide Turn ? Name Reflection Translation Rotation Reflection: When done on the y-axis, the x-sign changes. When done on the x-axis, the y-sign changes. Translation: can slide up or down, right or left or diagonally. • To right: Add number of units to x value • To left: Subtract number of units from x value • Up: Add number of units to the y value J -. Down: Subtract number of units from y value 1 RevC Triangles Proving Triangle Congruency Define • • Included Angle: Included Side: Postulates: • Postulate A: SSS If there exists a correspondence between the vertices of 2 triangles such that 3 sides of 1 triangle are congruent to 3 sides of other triangle, then the 2 triangles are congruent. • Postulate B: SAS If there exists a correspondence between the vertices of 2 triangles such that 2 sides and the included Z of 1 triangle are congruent to corresponding parts of the other triangle, then 2 triangles are congruent. • Postulate C: ASA If there exists a correspondence between the vertices of 2 triangles such that 2 Zs and the included side of I triangle are congruent to the corresponding parts of the other triangle, then 2 triangles are congruent. • Postulate D: AAS If there exists a correspondence between the vertices of 2 triangles such that 2 Zs and the nonincluded side of 1 triangle are congruent to the corresponding parts of the other triangle, then 2 triangles are congruent. Examples: 1. Use the tick marks for each pair of triangles, name the method, if any, that can be used to prove the triangles are congruent. C. b. d. a. g U 2. Name the additional congruent parts needed so that the triangles are congruent by the specified method. b. by AAS a. by SSS c. by ASA y E A Z 2 F A C R T RevC Triangles Congruency Proofs: 1. Given: AD CD, B is midpoint of AC Prove: tXABD ACBD Reason 1. Given 2. Given 3. 4. Reflexive Property 5. Statement 1. 3.ABECB 4. 5.AABDACBD 2. Given: T is midpoint of CH, AO Prove: ACAT AHOT Reason 1. Given 2. If point is midpoint, ÷ 2 3.Ifpointismidpoint, ÷2 4. 5. Statement 1. T is midpoint of CH, AO 2. 3. 4.ZCTAZOTH AHOT 5. ACAT ZH, T is the midpoint of AO 3. Given: rc Prove: ACAT AHOT Reason 1. 2. 3. If point is midpoint, ÷ 2 4.Vertica1/ 5. Statement 1.ZCZH 2. T is the midpoint of AO 3. 4. 5.ACATAHOT 4. Given /3 /6, KR PR, ZKRO Prove: AKRM APRO /PRM Reason 1. 2. 3. 4.AFD 5. if 2 / form straight / then sup 6. if 2 / form straight / then sup 7. sup ofE / are 8. 9. Statement 1./3Z6 2.KRPR 3.ZKRO/PRM 4. 5. 6. 7. 8./KRM/PRO 9. AKRM APRO 3 RevC _______L Triangles Congruency in Right Triangles ilL Postulate: • If the hypotenuse and the leg of I right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then triangles are congruent. HA Theorem: • If the hypotenuse and an acute angle of 1 right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then triangles are congruent. LA Theorem: • If 1 leg and an acute angle of 1 right triangle are congruent to corresponding leg and acute angle of another right triangle, then triangles are congruent. LL Theorem: • If legs of 1 right triangle are congruent to corresponding legs of another right triangle, then triangles are congruent. Examples: 5. Given:BC±AC,BD±AD,AC Prove: AACB AADB AD Reason 1. Given 2. 3. 4. 5.Defofl 6. 7. Reflexive 8. Statement 1. 2.BDIAD 3.ACAD 4. ZCisrightangle 5. 6. AACB and AADB are right triangles 7. 8.AACBAADB 6. Given:JK±KM,MLIIJK,JM KL Prove: AJKM ALMK Reason 1. 2. 3. 4. 5. 6. 7. 8. Reflexive 9. Statement 1.JK±KM 2.MLIIJK 3.JMKL 4. LJKM is right angle 5. ZJKM ZLMK 6. ZJMK ZLKM 7. AJKM and ALMK are right triangles 8. 9. AJKM ALMK 4 RevC Triangles CPCTC What is CPCTC? • Corresponding Parts of Congruent Triangles are Congruent Why would this be true? • Def of A says every pair of corresponding parts is When is it used? • Only after 2 A have been proven or stated to be . Cannot be used to prove A Define • Altitude: • Auxiliary Lines: • Bisect: • Median: Key Concepts/Theorems • All radii of a circle are congruent. Examples: 7. Given: Circle P Prove: AB CD Statement l.CircleP 2,PCPDPBPA 3. 4.ACPDABPA 5.ABCD Reason 1. 2. 3. Vertical angles 4. 5. 8. Given:AC AB,AEAD Prove: CE BD Statement 1. 2. 3.ZAXA 4.AAECAADB 5.CEBD Reason 1. Given 2. Given 3. 4. 5. CB 9. Given: Circle 0, ZT is comp to ZMOT, ZS is comp to ZPOS Prove: MO P0 Statement 1. 2. 3. 4. 5.ZMOTZPOS 6. APOS 7. AMOT 8,MOPO 5 Reason 1. Given 2. Given 3. Given 4. All radii 5. 6. Complements of 7. 8. Z are RevC Triangles 10. Given: AC BC, AD BD Prove: CD bisects /ACB Statement l.ACBC 2.ADBD 3.CDCD 4. 5./ACDZBCD 6. CD bisects ZACB AAB 11. Given: CD & BE are altitudes of AABC,AD AE Prove: DB EC Statement 1. CD and BE are altitudes_of AABC 2.ADAE 3. 4. 5. 6.rAzA 7. AAEB AADC 8.ABAC 9. Reason 1. 2. 3. 4. SSS 5. 6. Reason 1. Given 2. 3. if a line is an altitude, it is I and forms right angles. 4. if a line is an altitude, it is I and forms right angles 5. All right angles 6. 7. 8. 9. Subtraction Property Isosceles Triangles An1e-Side Theorems • If sides then angles: If 2 sides of a triangle are ,then the angles opposite those sides are • If angles then sides: If 2 angles of a triangle are • If 2 sides of a triangle are not congruent, then the angles opposite those sides are not congruent and the larger angle is opposite the longer side. • If 2 angles of a triangle are not congruent, then the sides opposite those angles are not congruent and the longer side is opposite the larger angle. , then the sides opposite those angles are Consequences of Angle-Side Theorems: • Is an equiangular triangle also equilateral? Why or why not? 6 RevC Triangles Examples: 12. Given: AC > AB, mZB + mZC < 180, mLB What are the restrictions on the value of x? = 6x-45, mLC=15+x 15 Statement 1.ZEZH 2.EFGH 3. 4. 5.DFDG 13. Given: ZE ZH, EF GH DG Prove: DF Er U Reason 1. 2. 3. if angles then sides 4. SAS 5. Right Angle Theorem Right-Ane Theorem If 2 angles are both supplementary and , then they are right angles. Examples: 14. Given: AB AC, BD CD Prove: AD is an altitude Statement 1. ABAC 2.BDCD 3. 4.txABD AACD 5.ZADB ZADC 6. XADB and ZADC are right angles 7. 8. AD is an altitude Reason 2. 3. Reflexive 4. 5. 6. 7. if form right angles, then I 8. Equidistance Theorems Define: • Distance: • Equidistant: • Perpendicular Bisector: Theorems/Postulates: • TPEEEDPB: if 2 points are each equidistant from the endpoints of a segment, then the 2 points determine the perpendicular bisector of that segment BD is I bis of AC 7 RevC _____________ Triangles • POPBTEE: if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. 15. Given: PQ is bisector of AB Prove: PAPB Statement 1.PQis±bisectorofAB 2. /PQA and ZPQB are right angles 3. 4. 5. PQ PQ 6. APQA APQB 7.PAPB Reason 1. 2. 3. All right angles are 4. Definition of bisect 5. 6. 7. Examples: 16. Given: Z1 /2, /3 Z4 Prove: AE Ibisector BD Statement 1./lZ2 2. AB AD 3. 4. 5. AE ±bisector BD 3. Given 4. if angles then sides 5. 17. Prove: The line joining the vertex of isosceles triangle to midpoint of base is perpendicular to base. Given: APIE is isosceles, S is midpoint of PE Prove: IS I PE Statement 1. APTE is isosceles 2.PIIE 3. 4.PSSE 5. ISIPE Reason 1. 2. 3. Given 4. 5. Reason 1. 18. Given: AB AD, BCCD. Prove: BE ED Statement 1.ABAD Reason 1. 2. Given 3. 4. 3.AEIbisBD 4. 8 RevC _________________________ _____________________________ ______________________ _________________therefore, ___________________________ ________________________________ ______LB ________________ ____________ ____________________ _______________ Triangles Indirect Proofs Define: • Indirect Proofs: method of proving where direct proof would be difficult to apply. Procedures for Indirect Proof 1. 2. 3. List the possibilities for the conclusion. a. Your conclusion is or is not true. Assume that the negation of the desired conclusion is true. a. So the OPPOSITE of the conclusion Write a “chain of reasons” until you reach an IMPOSSIBILITY or a CONTRADICTION. a. This will be a statement that either disputes a known theoremldefinitionlpostulate or your given information. 4. State that what you assumed to start was WRONG and that the desired conclusion then must be true. Examples: Paragraph Proofs I. Given LA LD, AB Prove: LB LE. DE, AC DF Proof: Either: ZB Assume:_____ LB Since, thus LA AABC ZE or ZE ZE Li), A13 DL ADEF by ASA and therefore AC Not possible since contradicts given that AC Therefore, assumption of/B LE — DF 1W is false and LB LE is true 2. Given: RS L PQ, PR QR Prove: RS does not bisect LPRQ Proof: Either: or Assume: Since, Not possible since contradicts and is true assumption of 9 is false RevC