Question

Please answer it with proper explanation....

Answer 1

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GIVEN:- ABCD is a trapezium in which AB//CD and AD=BC. 

Construction:- Draw a line through C parallel to DA intersecting AB produced at E.

Proof:- 

(1) AB//CD (Given) 

(2) AD//EC ( By construction )

(3) CE=AD (Opposite sides of parallelogram are equal) 

AD=BC (Given)

We know that,

Angle A + Angle E = 180°[ Co. Interior Angles] 

Angle E= 180°-Angle A 

Also, BC=CE 

Angle E = Angle CBE = 180° - Angle A 

Angle ABC= 180°-Angle CBE (ABE is a straight line)

Angle ABC= 180°-(180°-Angle A)

Angle ABC= 180°-180°+Angle A

Angle B= Angle A ..............................(1)

(2) Angle A+Angle D = Angle B + Angle C 180° (Angles on the same side of the transversal)

Angle A + Angle D = Angle B + Angle C

Angle A = Angle B [From eq (1)]

Hence, Angle D= Angle C 

(3) In ∆ ABC and ∆ABD 

AB=AB (Common)
Angle DBA = Angle CBA [From eq (1)]
AD=BC ( Given)

∆ABC IS equal and congruent to ∆ ABD (BY SAS CONGRUENCE CONDITION)

(4)Diagonal AC = Diagonal BC ( By CPCT AS ∆ABC is equal and congruent to ∆ ABD) 

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Answer 2

GIVEN:- ABCD is a trapezium in which AB//CD and AD=BC.

Construction:- Draw a line through C parallel to DA intersecting AB produced at E.

Proof:-

(1) AB//CD (Given)

(2) AD//EC ( By construction )

(3) CE=AD (Opposite sides of parallelogram are equal)

AD=BC (Given)

We know that,

Angle A + Angle E = 180°[ Co. Interior Angles]

Angle E= 180°-Angle A

Also, BC=CE

Angle E = Angle CBE = 180° - Angle A

Angle ABC= 180°-Angle CBE (ABE is a straight line)

Angle ABC= 180°-(180°-Angle A)

Angle ABC= 180°-180°+Angle A

Angle B= Angle A ..............................(1)

(2) Angle A+Angle D = Angle B + Angle C 180° (Angles on the same side of the transversal)

Angle A + Angle D = Angle B + Angle C

Angle A = Angle B [From eq (1)]

Hence, Angle D= Angle C

(3) In ∆ ABC and ∆ABD

AB=AB (Common)

Angle DBA = Angle CBA [From eq (1)]

AD=BC ( Given)

∆ABC IS equal and congruent to ∆ ABD (BY SAS CONGRUENCE CONDITION)

(4)Diagonal AC = Diagonal BC ( By CPCT AS ∆ABC is equal and congruent to ∆ ABD)

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