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Congruent Triangles: Identifying and Proving

Learn how to identify and prove congruent triangles using the properties of corresponding angles and sides. This lesson covers the Reflexive, Symmetric, and Transitive properties of congruent triangles.

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Congruent Triangles: Identifying and Proving

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  1. Warm-Up Exercises • Tell whether it is possible to draw each triangle. • Acute scalene triangle • Obtuse equilateral triangle • Right isosceles triangle • Scalene equiangular triangle • Right scalene triangle

  2. 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.

  3. Goal 1 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 PQR ABC BCA 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

  4. 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:

  5. 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

  6. 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

  7. Goal 1 Identifying Congruent Figures The Third Angles Theorem below follows from the Triangle Sum Theorem. 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.

  8. 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.

  9. Goal 2 Proving Triangles are Congruent  ,  , and  RP MN PQ NQ QR QM NQM PQR 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.

  10. 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

  11. 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 All vertical angles are congruent  AEB   DEC Given Definition of midpoint Definition of congruent triangles

  12. 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 In this lesson, 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. In upcoming lessons, you will learn more efficient ways of proving that triangles are congruent. The properties below will be useful in such proofs. THEOREM TheoremProperties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles

  13. Classwork: p. 205 #1-9 Assignment: pp. 206-9 #11-29 odd, 30-33, 35, 38, 39, 41-57 odd

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