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State whether the following statement is true or false:

If ABCD is a trapezium in which $A B ‖ C D$ and $A D = B C$ , then $∠ A = ∠ B$ .

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True

False

answer is A.

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Detailed Solution

Given that ABCD is a trapezium,

A B ‖ C D

and

A D = B C

Let E is a point on AB such that

A D ∥ C E

and

A E ∥ C D

In AECD both pairs of opposite sides are parallel.
Therefore, AECD is a parallelogram.

A D = C E

and

A D = B C

[

A B ∥ C D

]
Thus,

B C = C E

⇒ ∠ C E B = ∠ C B E

… (1) [Opposite angles of equal sides]
Now for

A D ∥ C E

and AE is transversal.

∠ A + ∠ C E B = 180 °

[sum of consecutive interior angles is 180°]

⇒ ∠ A = 180 ° − ∠ C E B

… (2)
Also, AE is a line

⇒ ∠ B + ∠ C B E = 180 °

[Linear angles]

⇒ ∠ B = 180 ° − ∠ C B E

… (3)
Therefore,

∠ A = ∠ B

[from (1), (2) and (3)]
Hence the given statement is true.
Hence, the correct option is 1.

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