Is the given statement true or false?

The parallelogram ABCD is such that $∠ A = 60 °$ and point P is the midpoint of the side CD. Then $∠ A P B = 90$ °.

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True

False

answer is A.

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Detailed Solution

Given is a parallelogram ABCD with

∠ A = 60 °

and point P is the midpoint of the side CD.
Question Image

In a parallelogram, opposite angles are equal.
So, we have

∠ A = ∠ C = 60 °

.
Now,

∠ A + ∠ B + ∠ C + ∠ D = 360 ° ∠ B + ∠ D = 360 ° − 60 ° + 60 ° 2 ∠ B = 240 ° ∠ B = ∠ D = 120 °

As AB||DP and AP is a transversal we have,

∠ A P D = ∠ P A B ∠ A P D = 60 ° 2 ∠ A P D = ∠ P A B = 30 °

... (i)
Consider AB||PC and BP is a transversal,

∠ A B P = ∠ C P B ∠ A B P = ∠ B 2 = 120 ° 2 ∠ A B P = ∠ C P B = 60 °

... (ii)
Now as DPC is a straight line,

∠ A P D + ∠ A P B + ∠ C P B = 180 ° 30 ° + ∠ A P B + 60 ° = 180 ° ∠ A P B = 90 °

Hence, the given statement is true.
So, option 1 is the correct choice.

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