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4.2 Congruence & Triangles

4.2 Congruence & Triangles. Geometry Taken from: http://www.dgelman.com/powerpoints/geometry/spitz/4.2Congruence%20and%20Triangles.ppt. Objectives:. Identify congruent figures and corresponding parts Prove that two triangles are congruent. Identifying congruent figures.

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4.2 Congruence & Triangles

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  1. 4.2 Congruence & Triangles Geometry Taken from: http://www.dgelman.com/powerpoints/geometry/spitz/4.2Congruence%20and%20Triangles.ppt.

  2. Objectives: • Identify congruent figures and corresponding parts • Prove that two triangles are congruent

  3. Identifying congruent figures • Two geometric figures are congruent if they have exactly the same size and shape. NOT CONGRUENT CONGRUENT

  4. Congruency • When two figures are congruent: • corresponding angles are congruent • corresponding sides are congruent.

  5. Corresponding angles A ≅ P B ≅ Q C ≅ R Corresponding Sides AB ≅ PQ BC ≅ QR CA ≅ RP Triangles B Q R A C P

  6. How do you write a congruence statement? • List the corresponding angles in the same order. You can write ∆ABC≅ ∆PQR, or ∆BCA≅ ∆QRP

  7. The congruent triangles. Write a congruence statement. Identify all parts of congruent corresponding parts. Ex. 1 Naming congruent parts

  8. 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 Ex. 1 Naming congruent parts

  9. In the diagram NPLM ≅ EFGH A. Find the value of x. You know that LM ≅ GH. So, LM = GH. 8 = 2x – 3 11 = 2x 11/2 = x Ex. 2 Using properties of congruent figures 8 m 110° (2x - 3) m (7y+9)° 72° 87° 10 m

  10. In the diagram NPLM ≅ EFGH B. Find the value of y You know that N ≅ E. So, mN = mE. 72°= (7y + 9)° 63 = 7y 9 = y Ex. 2 Using properties of congruent figures 8 m 110° (2x - 3) m (7y+9)° 72° 87° 10 m

  11. Third Angles Theorem • If any 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.

  12. Find the value of x. 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° = (2x + 30)° 30 = 2x 15 = x Ex. 3 Using the Third Angles Theorem (2x + 30)° 55° 65°

  13. Decide whether the triangles are congruent. Justify your reasoning. P ≅ N. By 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. Ex. 4 Proving Triangles are congruent 92° 92°

  14. 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 ≅ ∆JKL.

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