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ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other. If ∠ABC = 72°, then the measure of the ∠BCD is
- a)162°
- b)18°
- c)108°
- d)72°
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
ABCD is a cyclic trapezium whose sides AD and BC are parallel to each ...
As stated in above figure ABCD is a cyclic trapezium with Line AD and Line BC parallel to each other.
As per the properties of cyclic quadrilateral,
∠ABC + ∠ADC = 180°
72°+ ∠ADC = 180°
∠ADC = 108°
Also, Properties of parallel lines
∠ADC + ∠BCD = 180°
∠BCD = 180° - ∠ADC = 180° - 108° = 72°
∴ ∠ABC = ∠BCD
⇒ ∠BCD= 72°
As per the properties of cyclic quadrilateral,
∠ABC + ∠ADC = 180°
72°+ ∠ADC = 180°
∠ADC = 108°
Also, Properties of parallel lines
∠ADC + ∠BCD = 180°
∠BCD = 180° - ∠ADC = 180° - 108° = 72°
∴ ∠ABC = ∠BCD
⇒ ∠BCD= 72°
Most Upvoted Answer
ABCD is a cyclic trapezium whose sides AD and BC are parallel to each ...
We know that in a cyclic trapezium, the opposite angles are supplementary. Therefore, we have:
∠DAB + ∠BCD = 180°
We also know that AD and BC are parallel, so ∠DAB and ∠BCD are alternate angles and thus congruent. Let's call this angle x. Therefore:
x + x = 180°
2x = 180°
x = 90°
Now we know that ∠DAB and ∠BCD are both 90°. Since the sum of the angles in a quadrilateral is 360°, we can find the measure of the other two angles:
∠ABD + ∠BCD = 180°
∠ABD + 90° = 180°
∠ABD = 90°
Similarly,
∠ABC + ∠DAB = 180°
∠ABC + 90° = 180°
∠ABC = 90°
Therefore, ABCD is a rectangle with diagonals AC and BD intersecting at their midpoint.
∠DAB + ∠BCD = 180°
We also know that AD and BC are parallel, so ∠DAB and ∠BCD are alternate angles and thus congruent. Let's call this angle x. Therefore:
x + x = 180°
2x = 180°
x = 90°
Now we know that ∠DAB and ∠BCD are both 90°. Since the sum of the angles in a quadrilateral is 360°, we can find the measure of the other two angles:
∠ABD + ∠BCD = 180°
∠ABD + 90° = 180°
∠ABD = 90°
Similarly,
∠ABC + ∠DAB = 180°
∠ABC + 90° = 180°
∠ABC = 90°
Therefore, ABCD is a rectangle with diagonals AC and BD intersecting at their midpoint.
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ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other. If ∠ABC = 72°, then the measure of the ∠BCD isa)162°b)18°c)108°d)72°Correct answer is option 'D'. Can you explain this answer?
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