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Similarity of Triangles by SAS Criteria

Triangles I - Concepts

Similar Figures

Similarity of Triangles

Basic Proportionality Theorem and its Converse

Similarity of Triangles by AAA Criteria

Similarity of Triangles by AA Criteria

Similarity of Triangles by SSS Criteria

Similarity of Triangles by SAS Criteria

Class - 10th CBSE Subjects

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, ABC is a triangle in which AB = AC, D and E are points on the sides AB and AC respectively, such that AD = AE. Then $bigtriangleup ADEsim bigtriangleup ABC$ by _______________ criteria.
A. AAA	B. AA	C. SSS	D. SAS
Right Option : D
View Explanation

Question : 2

The above two triangles are similar by _________________ criteria.
A. SSS	B. SAS	C. AA	D. AAA
Right Option : B
View Explanation

Question : 3

Check whether the two triangles are similar or not. If so, write the similarity relation.
A. yes, SAS	B. yes, ASA	C. no	D. yes, SSS
Right Option : C
View Explanation