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Answers
The
closed figure formed by joining four non collinear points in an order is called
a quadrilateral.
·Trapezium:
A quadrilateral in
which one pair of opposite sides are parallel is called a trapezium.
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Given, ABCD is a trapezium in which AB||CD &
AD=BC
To Show:
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ΔABC ≅ ΔBAD
(iv) diagonal AC = diagonal BD
Construction: Draw a line through C parallel to DA
intersecting AB produced at E.
Proof:
i)
AB||CD(given)
AD||EC (by construction)
So ,ADCE is a parallelogram
CE = AD
(Opposite sides of a parallelogram)
AD = BC (Given)
We know that ,
∠A+∠E= 180°
[interior
angles on the same side of the transversal AE]
∠E= 180° - ∠A
Also, BC = CE
∠E = ∠CBE= 180° -∠A
∠ABC= 180° - ∠CBE
[ABE is a straight line]
∠ABC= 180° - (180°-∠A)
∠ABC= 180° - 180°+∠A
∠B= ∠A………(i)
(ii) ∠A + ∠D = ∠B + ∠C = 180°
(Angles on
the same side of transversal)
∠A + ∠D = ∠A + ∠C
(∠A = ∠B) from eq (i)
∠D = ∠C
(iii) In ΔABC and ΔBAD,
AB = AB (Common)
∠DBA = ∠CBA(from eq (i)
AD = BC (Given)
ΔABC ≅ ΔBAD
(by SAS congruence rule)
(iv) Diagonal AC = diagonal BD
(by CPCT as ΔABC ≅ ΔBAD)
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Hope this will help you...