Concepts, Theorems and Postulates that can be use to prove that triangles are congruent .

1. Concepts, Theorems and Postulates that can be use to prove that triangles are congruent.

2. Learning Target • I can identify and use reflexive, symmetric and transitive property in proving two triangles are congruent. • I can use theorems about line and angles in my proof.

3. Corresponding AnglesandCorresponding Sides

4. Goal 1 Learning Target Identifying Congruent Figures Two geometric figures are congruent if they have exactly the same size and shape. Each of the red figures is congruent to the other red figures. None of the blue figures is congruent to another blue figure.

5. Goal 1 Learning Target Identifying Congruent Figures For the triangles below, you can write , which reads “triangle ABCis congruent to triangle PQR.” The notation shows the congruence and the correspondence. AB  PQ BCA PQR ABC QRP BC  QR CA  RP There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write  . When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. Corresponding Angles Corresponding Sides  A P  B Q  C R

6. Example Naming Congruent Parts The diagram indicates that  . The congruent angles and sides are as follows. DEF RST   , , FD TR EF ST DE RS The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts. SOLUTION Angles:  DR, ES, FT Sides:

7. Example Using Properties of Congruent Figures You know that  . LM GH In the diagram, NPLM EFGH. Find the value of x. SOLUTION So, LM= GH. 8 = 2x– 3 11 = 2x 5.5 = x

8. Example Using Properties of Congruent Figures You know that  . LM GH In the diagram, NPLM EFGH. Find the value of x. Find the value of y. SOLUTION SOLUTION You know that NE. So, m N= m E. So, LM= GH. 72˚ = (7y+ 9)˚ 8 = 2x– 3 63 = 7y 11 = 2x 9 = y 5.5 = x

9. Theorems and Postulates on Congruent Angles (Transitive, Reflexive and Symmetry Theorem of Angle Congruence)

10. For any angle A, A  A REFLEXIVE If A B, then BA SYMMETRIC If A  B and B  C, then A  C TRANSITIVE CONGRUENCE OF ANGLES THEOREM THEOREM 2.2Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Here are some examples.

11. Transitive Property of Angle Congruence C B AB, AC GIVEN PROVE A BC Prove the Transitive Property of Congruence for angles. SOLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C.

12. Transitive Property of Angle Congruence 5 4 3 1 2 Statements Reasons A  B, Given AB, AC GIVEN PROVE B  C BC mA = m B Definition of congruent angles mB = m C Definition of congruent angles mA = m C Transitive property of equality A CDefinition of congruent angles

13. Using the Transitive Property m 3 = 40°, 12, 23 GIVEN m 1 = 40° PROVE 4 1 2 3 Statements Reasons m 3 = 40°, 1 2, Given 2 3 1 3Transitive property of Congruence m 1 = m 3 Definition of congruent angles m 1 = 40° Substitution property of equality This two-column proof uses the Transitive Property.

14. Right Angle Theorem

15. Proving Right Angle Congruence Theorem 1 and 2 are right angles GIVEN 1 2 PROVE THEOREM Right Angle Congruence Theorem All right angles are congruent. You can prove Right Angle CongruenceTheorem as shown.

16. Proving Right Angle Congruence Theorem 1 2 3 4 1 and 2 are right angles GIVEN Statements Reasons 1 2 PROVE 1 and 2 are right anglesGiven m 1 = 90°, m 2 = 90° Definition of right angles m 1 = m 2 Transitive property of equality 1  2Definition of congruent angles

17. Congruent Supplements Theorem (Supplementary Angles)

18. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2 1 3

19. 1 2 3 1 and 3 If m 1 + m 2 = 180° m 2 + m 3 = 180° 1  3 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. then

20. Congruent Complements Theorem (Complementary Angles)

21. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 5 6 4

22. 5 6 4 6 4 and If m 4 + m 5 = 90° m 5 + m 6 = 90° 4  6 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. then

23. Proving Congruent Supplements Theorem 1 2 Statements Reasons 1 and 2 are supplements Given 3 and 4 are supplements 1 4 m 1 + m 2 = 180° Definition of supplementary angles m 3 + m 4 = 180° 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE

24. Proving Congruent Supplements Theorem 3 5 4 m 1 + m 2 = Transitive property of equality m 3 + m 1 m 3 + m 4 m 1 = m 4 Definition of congruent angles m 1 + m 2 = Substitution property of equality 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

25. Proving Congruent Supplements Theorem 6 7 m 2 = m 3 Subtraction property of equality 2 3Definition of congruent angles 1 and 2 are supplements GIVEN 3 and 4 are supplements 1 4 2 3 PROVE Statements Reasons

26. Linear Pair Postulate

27. m 1 + m 2 = 180° PROPERTIES OF SPECIAL PAIRS OF ANGLES POSTULATE Linear Pair Postulate If two angles form a linear pair, then they are supplementary.

28. Proving Vertical Angle Theorem THEOREM Vertical Angles Theorem Vertical angles are congruent 1 3, 2 4

29. Proving Vertical Angle Theorem 5 and 6 are a linear pair, GIVEN 6 and 7 are a linear pair 5 7 1 2 3 PROVE Statements Reasons 5 and 6 are a linear pair,Given 6 and 7 are a linear pair 5 and 6 are supplementary,Linear Pair Postulate 6 and 7 are supplementary 5 7 Congruent Supplements Theorem

30. Third Angles Theorem

31. Goal 1 The Third Angles Theorem below follows from the Triangle Sum Theorem. THEOREM Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. IfADandBE,thenCF.

32. Example Using the Third Angles Theorem Find the value of x. SOLUTION In the diagram, NR and LS. From the Third Angles Theorem, you know that MT. So, m M = m T. From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚. m M= m T Third Angles Theorem 60˚ = (2x+ 30)˚ Substitute. 30 = 2x Subtract 30 from each side. 15 = x Divide each side by 2.

33. Goal 2 Learning Target Proving Triangles are Congruent  ,  , and  RP MN PQ NQ QR QM PQR NQM So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles,  . Decide whether the triangles are congruent. Justify your reasoning. SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. Because P and N have the same measures, P N. By the Vertical Angles Theorem, you know that PQR NQM. By the Third Angles Theorem, R M.

34. Example Proving Two Triangles are Congruent Prove that  . AEB DEC A B E || , AB DC D C E is the midpoint of BC and AD. GIVEN  . PROVE AEB DEC Plan for Proof Use the fact that AEB and  DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.  , AB DC

35. Example Proving Two Triangles are Congruent Prove that  . AEB DEC  EAB   EDC, A B  ABE   DCE E D C  || , AB DC AB DC E is the midpoint of AD, E is the midpoint of BC ,  AE BE CE DE  DEC AEB SOLUTION Given Alternate Interior Angles Theorem Vertical Angles Theorem  AEB   DEC Given Definition of midpoint Definition of congruent triangles

36. Goal 2 Proving Triangles are Congruent E B D A F C DEF DEF ABC DEF ABC ABC DEF JKL ABC JKL If  , then  . L If  and  , then . J K You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. THEOREM Theorem 4.4Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles

37. then If 1.ABDE 4.AD 2.BCEF 5. BE ABCDEF 3.ACDF 6.CF SSS AND SASCONGRUENCE POSTULATES If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Sides are congruent Angles are congruent Triangles are congruent and

38. S S S Side MNQR then MNPQRS Side NPRS Side PMSQ SSS AND SASCONGRUENCE POSTULATES POSTULATE POSTULATE:Side -Side -Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent. If

39. Using the SSS Congruence Postulate Prove that PQWTSW. The marks on the diagram show that PQTS, PWTW, andQWSW. SOLUTION Paragraph Proof So by the SSS Congruence Postulate, you know that PQW TSW.

40. POSTULATE Side PQWX A S S then PQSWXY Angle QX Side QSXY SSS AND SASCONGRUENCE POSTULATES POSTULATE:Side-Angle-Side (SAS) 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. If

41. Using the SAS Congruence Postulate Prove that AEBDEC. 1 2 1 2 Statements Reasons AE  DE, BE  CE Given 1  2Vertical Angles Theorem 3 AEBDEC SAS Congruence Postulate

42. Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make DRAcongruent to DRG. You design the window so that DRAG and RARG. D A G R GIVEN DRAG RARG DRADRG PROVE MODELING A REAL-LIFE SITUATION Can you conclude that DRADRG? SOLUTION

43. Proving Triangles Congruent GIVEN RARG DRADRG PROVE 1 2 6 3 4 5 Statements Reasons Given DRAG If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. Right Angle Congruence Theorem DRADRG DRAG Given RARG DRDR Reflexive Property of Congruence SAS Congruence Postulate DRADRG D A R G

44. Congruent Triangles in a Coordinate Plane AC FH ABFG Use the SSS Congruence Postulate to show that ABCFGH. SOLUTION AC = 3 and FH= 3 AB = 5 and FG= 5

45. Congruent Triangles in a Coordinate Plane d = (x2 – x1 )2+ (y2 – y1 )2 d = (x2 – x1 )2+ (y2 – y1 )2 BC = (–4 – (–7))2+ (5– 0)2 GH = (6 – 1)2+ (5– 2)2 = 32+ 52 = 52+ 32 = 34 = 34 Use the distance formula to find lengths BC and GH.

46. Congruent Triangles in a Coordinate Plane BCGH BC = 34 and GH= 34 All three pairs of corresponding sides are congruent, ABCFGH by the SSS Congruence Postulate.