Similarity of Triangles by SAS Criteria


 
 
Concept Explanation
 

Similarity of Triangles by SAS Criteria

Similarity of Triangles by SAS Criteria:

Theorem:  If in two triangles one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.

Given in large Delta ABC and large Delta DEF

          large frac{AB}{DE}=frac{AC}{DF}   and  large angle A=angle D

To Prove   large Delta ABCsim Delta DEF

Construction: Cut DX equal to AB and DY equal to AC. Join XY

Proof: It has been given

    large frac{AB}{DE}=frac{AC}{DF}

Replacing Ab with DX and AC with DY

   large frac{DX}{DE}=frac{DY}{DF}

Taking Reciprocal and subtracting 1 from both sides

large frac{DE}{DX}-1=frac{DF}{DY}-1Rightarrow frac{DE-DX}{DX}=frac{DF-DY}{DY}

large frac{XE}{DX}=frac{YF}{DY}Rightarrow XYparallel EF     [ Converse of basic proportionality theorem ]

large angle X= angle E;;and;;angle Y=angle F   [Corresponding angles]   .......................(1)

large In; Delta ABC ;and; Delta DXY

     AB = DX                        [By Construction ]

      AC = DY                        [By Construction ]

       large angle A=angle D        [Given]

large Delta ABC cong ; Delta DXY

large angle B= angle X;;and;;angle C=angle Y                     ........................(2)

From Eq (1) and (2)

  large angle B= angle E;;and;;angle C=angle F

Therefore

large Delta ABCsim Delta DEF           [By AAA Similarity Criterion]

Illustration: If AD and PM are medians of triangles ABC and PQR respectively where large Delta ABC simDelta PQR, Prove that large frac{AB}{PQ}=frac{AD}{PM}

Solution: Given large Delta ABC simDelta PQR,

large frac{AB}{PQ}=frac{BC}{QR}=frac{AC}{PR}                                      .............................(1)

large angle A=angle P, angle B=angle Q ;and;angle C=angle R

As AD and PM are the medians so

large BD=CD=frac{1}{2}BC;;and;;QM=MR=frac{1}{2}QR  

BC= 2BD=2CD      and QR= 2 QM= 2MR                     ......................(2)

From Eq 1

large frac{AB}{PQ}=frac{BC}{QR} Rightarrow frac{AB}{PQ}=frac{2BD}{2QM}=frac{BD}{QM}                ......................(3)  

large In ;Delta ABM ;and;Delta PQM

   large frac{AB}{PQ}=frac{BD}{QM}                          [From Equation (3)

large angle B=angle Q                                 [Proved Above]

large Rightarrow Delta ABD simDelta PQM                    [By SAS Similarity Criterion ]

   Thus large frac{AB}{PQ}=frac{AD}{PM}

Hence Proved

Sample Questions
(More Questions for each concept available in Login)
Question : 1

In the following figure, AB || QR. Find the length of PB.

Right Option : D
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Explanation
Question : 2

In the following figure, XY || BC, Find the length of XY.

Right Option : C
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Explanation
Question : 3

The above two triangles are similar by _________________ criteria.

Right Option : B
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Explanation
 
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