EFGH is a square and TGH is an equilateral triangle . prove TE = TF , angle TFG
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Answer:
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Here, PQRS is a square
⇒ PQ=QR=RS=SP [ Sides of square are equal ] ------ ( 1 )
⇒ ∠SPQ=∠PQR=∠QRS=∠RSP=90
o
[ Angles of square ]
△SRT is an equilateral triangle.
⇒ SR=RT=TS [ Sides of equilateral triangle are equal ] ----- ( 2 )
⇒ ∠TSR=∠SRT=∠RTS=60
o
From ( 1 ) and ( 2 ),
⇒ PQ=QR=SP=SR=RT=TS ----- ( 3 )
⇒ ∠TSP=∠TSR+∠RSP=60
o
+90
o
=150
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⇒ ∠TRQ=∠TRS+∠SRQ=60
o
+90
o
=150
o
⇒ ∠TSP=∠TRQ=150
o
----- ( 4 )
Now, in △TSP and △TRQ
⇒ TS=TR [ From ( 3 ) ]
⇒ ∠TSP=∠TRQ [ From ( 4 ) ]
⇒ SP=RQ [ From ( 3 ) ]
∴ △TSP≅△TRQ [ By SAS congruence rule ]
⇒ PT=QT [ CPCT ] ---- Hence proved
Consider △TQR,
⇒ QR=TR [ From ( 3 ) ]
∴ △TQR is an isosceles triangle.
⇒ ∠QTR=∠TQR [ Angles opposite to equal sides ]
Now, a sum of angles in a triangle is equal to 180
o
.
⇒ ∠QTR+∠TQR+∠TRQ=180
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⇒ 2∠TQR+150
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=180
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[ From ( 4 ) ]
⇒ 2∠TQR=180
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−150
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⇒ 2∠TQR=30
o
⇒ ∠TQR=15
o
---- Hence proved