4.2 Congruence and Triangles

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  - Congruence of Figures - Corresponding Parts - Third Angle Theorem
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  • 1. Geometry - 4.2Congruence & Triangles
  • 2. Congruent, Corresponding Angles/Sides Two figures are congruent when their corresponding sides and corresponding angles are congruent.Corresponding Angles < A ≅< P < B ≅< Q ΔABC ≅ ΔPQR < C ≅< RCorresponding Sides AB ≅ PQ There is more than one way to write a BC ≅ QR congruence statement, but the you must list CA ≅ RP the corresponding angles in the same order.
  • 3. Naming Congruent Parts Write a congruence statement for the triangles below. Identify all pairs of congruent parts. ΔABC ≅ ΔZXYCorresponding Angles Corresponding Sides < A ≅< Z XY ≅ BC < B ≅< X YZ ≅ AC < C ≅< Y XZ ≅ AB
  • 4. Identify Corresponding Congruent PartsShow that the polygons arecongruent by identifying all ofthe congruent correspondingparts. Then write a congruencestatement.Angles:Sides:Answer: All corresponding parts of the two polygons arecongruent. Therefore, ABCDE ≅ RTPSQ.
  • 5. Third Angle ThmThird Angle Theorem. - If two angles of one triangle arecongruent to two angles of another triangle, then the thirdangles are also congruent. If < A ≅< D and < B ≅< E then, < C ≅< F
  • 6. Properties of Congruent Triangles Reflexive Property of Congruent Triangles ΔABC ≅ ΔABC Symmetric Property of Congruent TrianglesIf ΔABC ≅ ΔDEF, then ΔDEF ≅ ΔABC Transitive Property of Congruent TrianglesIf ΔABC ≅ ΔDEF and ΔGHI ≅ ΔDEF,then ΔABC ≅ ΔGHI
  • 7. Proof of Third Angle ThmGiven: <A ≅ <D, <B ≅ <EProve: <C ≅ <F1. <A ≅ <D, <B ≅ <E 1. Given2. m<A = m<D, m<B = m<E 2. Def’n of Congruent Angles3. m<A + m<B + m<C = 180 3. Triangle Sum Theorem4. m<D + m<E + m<F = 180 4. Triangle Sum theorem5. m<A + m<B + m<C = m<D + m<E + m<F 5. Transitive Property6. m<D + m<E + m<C = m<D + m<E + m<F 6. Substitution Property7. m<C = m<F 7. Subtraction Property8. <C ≅ <F 8. Def’n of Congruent Angles
  • 8. Using the Third Angle Thm.Find the value of x.22 + 87 + m < A = 180109 + m < A = 180m < A = 71m<D=m< A4 x + 15 = 714 x = 56x = 14
  • 9. Determining Triangle CongruencyDecide whether the triangles are congruent. Justify your reasoning.From the diagram all correspondingsides are congruent and that <F and<H are congruent.<EGF and <HGJ are congruentbecause of Vertical angles. Since all of the corresponding sides and angles are congruent,<E and <J are congruent because ofthe third angle theorem ΔEFG ≅ ΔHJG
  • 10. Using Properties of Congruent FiguresIn the diagram, ABCD ≅ KJHL 4x − 3 = 9a) Find the value of x. 4 x = 12b) Find the value of y. x=3 5 y − 12 = 113 5 y = 125 y = 25
  • 11. Use Corresponding Parts of Congruent TrianglesIn the diagram, ΔITP ≅ ΔNGO. Find the values of x and y. x – 2y = 7.5 x – 2(9) = 7.5 x – 18 = 7.5 ∠O ≅ ∠P x = 25.5 6y – 14 = 40 Answer: x = 25.5, y = 9 6y = 54 y= 9
  • 12. In the diagram, ΔFHJ ≅ ΔHFG. Find the values of x and y. A. x = 4.5, y = 2.75 B. x = 2.75, y = 4.5 C. x = 1.8, y = 19 D. x = 4.5, y = 5.5
  • 13. Prove: ΔLMN ≅ ΔPON Proof: Statements Reasons1. 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles Theorem3. ∠M ≅ 3. Third Angles Theorem ∠O4. ΔLMN 4. Def of Congruent Triangles ≅
  • 14. Proving Two Triangles CongruentGiven: MN ≅ QP, MN || PQO is the midpt of MQ and PNProve: ΔMNO ≅ ΔQPO• 1) O is the midpoint of MQ and PN • 1) Given MN ≅ QP, MN || PQ• 2) < OMN ≅< OQP, < MNO ≅< QPO • 2) Alt. Int. <‘s Thm.• 3) < MON ≅< QOP • 3) Vertical <‘s• 4) MO ≅ QO, PO ≅ NO • 4) Def of Midpoint• 5) ΔMNO ≅ ΔQPO • 5) Def of Congruent Tri<‘s
  • 15. Practice Problems• Textbook p206: 14-32 even, 35
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