Question

In the figure, $PQ=QR=RS=SP$. Then triangles are congruent by the postulate

• A. SSS
• B. SAS
• C. ASA
• D. All of these

Answer 1

Answer: (D)

All of these

$\Rightarrow$ $PQ=OR=RS=SP$ [ Given ]

$\Rightarrow$ $\angle SPR=\angle SRP$ [ Base angles of equal sides also equal ]

$\Rightarrow$ $\angle PRQ=\angle RPQ$ [ Base angles of equal sides also equal ]

$\therefore$ $\angle SPR=\angle SRP=\angle PRQ=\angle RPQ$ (because geometry is a Rhombus)

In $△PRS$ and $△PRQ$

$\Rightarrow$ $PR=PR$ [ Common side ]

$\Rightarrow$ $RS=QR$ [ Given ]

$\Rightarrow$ $SP=PQ$ [ Given ]

$\therefore$ $△PRS\cong △PRQ$ [ By $SSS$ congruence rule ]

Now,

$\Rightarrow$ $SR=PQ$

$\Rightarrow$ $\angle SRP=\angle RPQ$ [ From ( 1 ) ]

$\Rightarrow$ $PR=RP$ [ Common side ]

$\therefore$ $△PRS\cong △PRQ$ [ By $SAS$ congruence rule ]

Now,

$\Rightarrow$ $\angle SRP=\angle RPQ$ [ From ( 1 ) ]

$\Rightarrow$ $PR=RP$ [ Common side ]

$\Rightarrow$ $\angle SPR=\angle PRQ$ [ From ( 1 ) ]

$\therefore$ $△PRS\cong △PRQ$ [ By $ASA$ congruence rule ]