Published on January 16th, 2013 In category Geometry | Maths

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Similarity and congruency of triangles

Mid-Point-TheoremSimilarity and congruency of triangles are important concepts as lots of questions are asked in competitive exams are based in these topics. Both these topics are governed by tests like SAS test, SSS test, AAS test RHS test in case of congruency of triangles and AAA test, AA test or SAS test in case the triangles are similar. Lets read this topic and latter on we will also solve some problems based on these concepts

So first we will start with understanding what the congruency and similarity of triangle means

Two triangles are said to be congruent when corresponding angles and sides are EQUAL. please note emphasis is on word equal. Corresponding sides  means sides opp. to angle similarly corresponding angles means angles opposite to equal sides.

To test if the two triangles are congruent following are the Tests of Congruency

  1. SAS Test : This test means Side Angle Side are respectively equal in two triangles. The angle included is angle between the two sides
  2. SSS Test : In this test all the 3 sides (SSS) are respectively equal to their corresponding sides in other triangle
  3. AAS Test: Here the side included between two angles are equal in correspondence of the two triangles
  4. RHS Test: The one side and hypotenuse of one triangle is equal to other triangle.

Two triangles are said to be similar when angles are equal but sides are only proportional.

To test if the two triangles are similar following are the Tests of similarity

  1. AAA test-  All the angles of one triangle are respectively equal to another triangle
  2. AA test- Two included angles are equal are respectively equal to two angles of the other triangles
  3. SAS test- The 2 sides are proportional and included angle is equal. 

 Some important theorems

As a rule states if 3 or more parallel lines are cut by 1 or more line in transverse, then intercepts formed are proportional to each other. This can be truely represented in a triangle if we draw a line such that it cuts 2 sides at 2 points such that it is parallel to third side, then using the above rule the segments should be proportional.

For example as shown in the diagram If AB = AE, then AC = AD 

 Midpoint Theorem

IMP.  The segment joining the midpoints of the two sides of the triangle is equal and half of the third side.


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