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Congruent triangles|Class 9 Maths notes






What is Congruence

  • In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
  • A combination of rigid motions, namely a translation, a rotation, and a reflection is also permitted in Congruence. Translation means sliding,Rotation means turning and Reflection means Flipping. If two figures can be overlapped using translation,rotation or reflection then they are congurence
  • It has come from Latin word congruere "agree, correspond with"

What is the difference between Congurence and Similarity?

Two Figures are similar if the objects differ in size but not in shape while in congurence size and shape remains same

Why do we need to Learn it?

  • We need to learn congurence in order to identify and describe many congurent object in real life. It may be some sculpture, some old monuments,
  • In Pyramid, all the three triangular surface are congurent.AROUND 2600 B.C., construction of the Great Pyramid of Khufu began. It took the ancient Egyptians about 30 years to transform 6.5 million tons of stone into a pyramid with a square base and four congruent triangular faces
  • We need several identical shirts button ,various factory part and many numerous objects. All these need the concept of Congurence


      Congruence

      Two Geometric figure are said to be congruence if they are exactly same size and shape
      • Symbol used is ≅
      • Two angles are congruent if they are equal
      • Two circle are congruent if they have equal radii
      • Two squares are congruent if the sides are equal
      • Two line segments are congruent if they have equal length

      Triangle Congruence

      • Two triangles are congruent if three sides and three angles of one triangle is congruent to the corresponding sides and angles of the other
        Triangle congurence
      • Corresponding sides are equal AB=DE , BC=EF ,AC=DF
      • Corresponding angles are equal Triangle congurence
      • We write this as
        ABC ≅ DEF
      • The above six equalities are between the corresponding parts of the two congruent triangles. In short form this is called C.P.C.T
      • We should keep the letters in correct order on both sides

      Properties of Congurent triangles


      1) Reflexive Property of Congurent Triangle: Every Triangle is congurent to itself
      ABC ≅ ABC
      2) Commutative or symmetric property of Congurent Triangle:
      if ABC ≅ DEF then DEF ≅ ABC

      3) Transitive Property Of congurent Triangle
      if ABC ≅ DEF and DEF ≅ LKM then ABC ≅ LKM

      Since Congurence relation satisfy Reflexive ,symmetric and Transitive ,it is an equivalence relation

      Different Criterion for Congruence of the triangles

      SAS Congurence (AXIOM)

      Congurent triangles

      ASA Congurence (Theorem)

      Congurent triangles

      AAS Congurence (Theorem)

      AAS Triangle congurence

      SSS Congurence (Theorem)

      SSS Triangle congurence

      RHS Congurence (Theorem)

      RHS Triangle congurence
      Caveat
      AAA( Angle Angle Angle) is not the right condition to Prove congurence.

      How to Prove the congurence of Two Triangle

      1) We have already studied that two triangles are congurent when all the sides and all the angles are equal. But we dont need prove all these while solving the Problem
      2) We just need to prove the congurence using the different criterio like SSS,ASA,SAS,RHS,AAS
      3) Dont use AAA
      4) Use the theorem learn in previous Geometry chapter like vertically opposite angles,alternate interior angles,corresonding angles
      5)Write down the corresponding angles and corresponding sides carefully
      6) We need to be careful with the labelling when our Triangles are in different positions

      Inequality of Triangle

      1) In a triangle angle opposite to longer side is larger
      Congurent triangles
      2) In a triangle side opposite to larger angle is larger
      3) The sum of any two sides of the triangle is greater than the third side
      In triangle ABC
      AB +BC > AC
      4) Difference of any two sides of triangle is less than the third side
      AC-AB < BC where AC > AB

      Important defination for Triangles

      Median

      A line Segment joining the corner of the triangle to the mid point of the opposite side of the triangle. All the median of the triangle passes through the same point

      Centroid

      Point of intersection of the three median of the triangle is called the centroid of the triangle. The centroid divides each of the median in the ration of 2:1

      Altitude

      A line Segment from the corner of the triangle and perpendicular to the opposite side of the triangle. All the altitude of the triangle passes through the same point

      Ortho center

      Point of intersection of the three altitude of the triangle is called the orthocenter of the triangle.

      Incenter

      All the angle bisector of the triangle passes through same point .It is called incenter.

      Circumcenter

      The perpendicular bisector of the sides of the triangles passes through same point. It is called circumcenter

      Solved Example

      Congurent triangles What is the measure of the angles P,M,R,N?
      Write down the corresponding congurent sides?
      Solution:
      By Congurence, All the corresponding angles and corresponding sides are equal
      LMN ≅ PQR
      Steps to find the All the corresponding angles and corresponding sides
      1) For angles, The First alphabet on right hand side corresponds to First alphabet on Left hand side,Similary second and third alphabet
      So ∠ L=∠ P
      ∠M=∠ Q
      ∠ N=∠ R
      Now ∠ L=1050
      ∠ Q=45, So ∠ M=45 0
      ∠P=1050
      Now ∠ R can be found from the theorem
      "Sum of the all the angles in a traingle is 1800"
      So ∠ R=180-105-45=300
      ∠ N=30 0
      2) For sides,The first two alphabet on right hand side corresponds to First two alphabet on Left hand side,Similary 2,3 and 3,1
      LM=PQ
      MN=QR
      LN=PR

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