# Concepts Theorems and Postulates that can be use

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

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. Goal 1

Learning Target 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. For the triangles below, you can write ABC PQR , which reads “triangle ABC is congruent to triangle PQR. ” The notation shows the congruence and the correspondence. Corresponding Angles Corresponding Sides A P B Q C R AB PQ 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 BCA QRP. Goal 1

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

Using Properties of Congruent Figures In the diagram, NPLM EFGH. Find the value of x. SOLUTION You know that LM GH. 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. Find the value of y. SOLUTION You know that LM GH. You know that N E. So, LM = GH. So, m N = m E. 8 = 2 x – 3 11 = 2 x 5. 5 = x 72˚ = (7 y + 9)˚ 63 = 7 y 9=y Example

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

CONGRUENCE OF ANGLES THEOREM 2. 2 Properties of Angle Congruence Angle congruence is r ef lex ive, sy mme tric, and transitive. Here are some examples. REFLEX IVE For any angle A, SYMMETRIC If A B, then TRANSITIVE If A B and B C, then A A B A A C

Transitive Property of Angle Congruence 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. GIVEN A B, B C PROVE A C C B A

Transitive Property of Angle Congruence A B, B C GIVEN Statements 1 A B, B C PROVE A C Reasons Given 2 m A=m B Definition of congruent angles 3 m B=m C Definition of congruent angles 4 m A=m C Transitive property of equality 5 A C Definition of congruent angles

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

Right Angle Theorem

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

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

Congruent Supplements Theorem (Supplementary Angles)

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. 1 2 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. 1 2 3 If m m 1+m 2+m 1 3 2 = 180° 3 = 180° and then 1

Congruent Complements Theorem (Complementary Angles)

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 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. 5 4 If m m 4+m 5+m 4 6 5 = 90° and 6 = 90° then 4 6

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

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

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

Linear Pair Postulate

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

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

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

Third Angles Theorem

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. Goal 1

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

Learning Target Proving Triangles are Congruent 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. RP MN, PQ NQ , and QR QM 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. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, PQR NQM. Goal 2

Proving Two Triangles are Congruent Prove that AEB DEC. A B E C GIVEN PROVE D AB || DC , AB DC, E is the midpoint of BC and AD. 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. Example

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

Proving Triangles are Congruent 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 B Theorem 4. 4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles If ABC DEF , then DEF ABC. Transitive Property of Congruent Triangles If ABC DEF and DEF JKL , then ABC A C E D F L JKL. J K Goal 2

SSS AND SAS CONGRUENCE POSTULATES If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. If Sides are congruent and Angles are congruent 1. AB DE 4. A D 2. BC EF 5. B E 3. AC DF 6. C F then Triangles are congruent ABC DEF

SSS AND SAS CONGRUENCE POSTULATES POSTULATE: Side - Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. S If Side MN QR S Side NP RS S Side PM SQ then MNP QRS

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

SSS AND SAS CONGRUENCE 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 S Side A Angle S Side PQ WX Q X QS XY then PQS WXY

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

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

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

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

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

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

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