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

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.

Corresponding Angles and Corresponding Sides

Goal 1 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. Learning Target

Goal 1 Identifying Congruent Figures 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 AnglesCorresponding Sides  A   P A   P  B   Q  C   R BC  QR RP CA  AB  PQ For the triangles below, you can write , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence. ABCPQR 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 . BCAQRP Learning Target

Example Naming Congruent Parts The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts. SOLUTION Angles: Sides:  D   R,  E   S,  F   T ,  RSDE TRFD STEF The diagram indicates that . The congruent angles and sides are as follows. DEFRST

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

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

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

C ONGRUENCE OF A NGLES THEOREM THEOREM 2.2 Properties of Angle Congruence Angle congruence is r ef lex ive, sy mme tric, and transitive. Here are some examples. TRANSITIVE IfA  BandB  C, then A  C SYMMETRIC If A  B, then B  A REFLEX IVE For any angle A, A  A

Transitive Property of Angle Congruence Prove the Transitive Property of Congruence for angles. S OLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C. GIVEN A B, PROVE A CA C A B C B CB C

Transitive Property of Angle Congruence GIVEN A B, B CB C PROVE A CA C StatementsReasons mA = mB Definition of congruent angles 5 A  C Definition of congruent angles A  B,Given B  C mB = mC Definition of congruent angles mA = mC Transitive property of equality

Using the Transitive Property This two-column proof uses the Transitive Property. StatementsReasons m1 = m3 Definition of congruent angles GIVEN m3 = 40°,12,23 PROVE m1 = 40° 1 m1 = 40° Substitution property of equality 13 Transitive property of Congruence Givenm3 = 40°,12, 2323

Right Angle Theorem

Proving Right Angle Congruence Theorem THEOREM Right Angle Congruence Theorem All right angles are congruent. You can prove Right Angle CongruenceTheorem as shown. GIVEN 1 and2 are right angles PROVE 1212

Proving Right Angle Congruence Theorem StatementsReasons m1 = 90°, m2 = 90° Definition of right angles m1 = m2 Transitive property of equality 1  2 Definition of congruent angles GIVEN 1 and2 are right angles PROVE and2 are right angles Given

Congruent Supplements Theorem (Supplementary Angles)

P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent

P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent If m1 + m2 = 180° m2 + m3 = 180° and 1 then 1  3

Congruent Complements Theorem (Complementary Angles)

P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent

P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 4 If m4 + m5 = 90° m5 + m6 = 90° and then 4 

Proving Congruent Supplements Theorem StatementsReasons 1 2 GIVEN 1 and2 are supplements PROVE and4 are supplements and2 are supplementsGiven 3 and4 are supplements 1  4 m1 + m2 = 180° Definition of supplementary angles m3 + m4 = 180°

Proving Congruent Supplements Theorem StatementsReasons 3 GIVEN 1 and2 are supplements PROVE and4 are supplements m1 + m2 = Substitution property of equality m3 + m1 m1 + m2 = Transitive property of equality m3 + m4 m1 = m4 Definition of congruent angles

Proving Congruent Supplements Theorem StatementsReasons GIVEN 1 and2 are supplements PROVE and4 are supplements m2 = m3 Subtraction property of equality 23 Definition of congruent angles

Linear Pair Postulate

POSTULATE Linear Pair Postulate If two angles for m a linear pair, then they are supplementary. m1 + m2 = 180° P ROPERTIES OF S PECIAL P AIRS OF A NGLES

Proving Vertical Angle Theorem THEOREM Vertical Angles Theorem Vertical angles are congruent 1 3,24

Proving Vertical Angle Theorem PROVE 5757 GIVEN 5 and6 are a linear pair, 6 and7 are a linear pair StatementsReasons 5 and6 are a linear pair, Given 6 and7 are a linear pair 5 and6 are supplementary, Linear Pair Postulate 6 and7 are supplementary 5 7 Congruent Supplements Theorem

Third Angles Theorem

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   D and  B   E, then  C   F.

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 60 ˚ = (2 x + 30) ˚ 30 = 2 x 15 = x Third Angles Theorem Substitute. Subtract 30 from each side. Divide each side by 2.

Goal 2 SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. , NQPQ , MNRP  QMQR and 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. Decide whether the triangles are congruent. Justify your reasoning. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, . PQRNQM Proving Triangles are Congruent Learning Target

Example Proving Two Triangles are Congruent AB C D E ||, DCAB , DCAB E is the midpoint of BC and AD. 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. GIVEN PROVE . AEBDEC Prove that . AEBDEC

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

Goal 2 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 4.4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles D E F A B C J K L Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles If , then . ABCDEF ABC If  and , then . JKLABCDEF ABCJKL Proving Triangles are Congruent

SSS AND SAS C ONGRUENCE P OSTULATES If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. and then If Sides are congruent 1. AB DE 2. BC EF 3. AC DF Angles are congruent 4. AD 5. BE 6. CF Triangles are congruent  ABC  DEF

SSS AND SAS C ONGRUENCE P OSTULATES POSTULATE POSTULATE: Side - Side - Side (SSS) Congruence Postulate Side MNQR Side PMSQ Side NPRS If If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. then  MNP  QRS S S S

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

POSTULATE SSS AND SAS C ONGRUENCE P OSTULATES POSTULATE: Side-Angle-Side (SAS) Congruence Postulate Side PQWX Side QSXY then  PQS  WXY Angle QX If 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. ASS

1 Using the SAS Congruence Postulate Prove that  AEB  DEC. 2 3  AEB   DEC SAS Congruence Postulate 21 AE  DE, BE  CEGiven 1  2Vertical Angles Theorem StatementsReasons

D G A R Proving Triangles Congruent M ODELING A R EAL- L IFE S ITUATION PROVE  DRA  DRG S OLUTION A RCHITECTURE You are designing the window shown in the drawing. You want to make  DRA congruent to  DRG. You design the window so that DR AG and RA  RG. Can you conclude that  DRA   DRG ? GIVEN DR AG RA RG

SAS Congruence Postulate  DRA   DRG 1 Proving Triangles Congruent Given DR AG If 2 lines are, then they form 4 right angles. DRA and DRG are right angles. Right Angle Congruence Theorem DRA  DRG Given RA  RG Reflexive Property of CongruenceDR  DR StatementsReasons D GAR GIVEN PROVE  DRA  DRG DR AG RA RG

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

Congruent Triangles in a Coordinate Plane d = (x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2 = = 34 BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2 d = (x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2 = = 34 GH = (6 – 1) 2 + (5 – 2 ) 2 Use the distance formula to find lengths BC and GH.

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