Question

"Question 12 ABCD is a trapezium in which AB || CD and AD = BC (see the given figure). Show that (i) ∠A = ∠B (ii) ∠C = ∠D (iii) ΔABC ≅ ΔBAD (iv) diagonal AC = diagonal BD [Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

Class 9 - Math - Quadrilaterals Page 147"

Answer 1

Quadrilateral:

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...

Answer 2

Answer:

hi mate

here is your answer

hope it will help you....