Raxit85
Is one of the angles of the quadrilateral ABCD equal to 90 degrees?
(1) ABCD is a parallelogram
(2) One of the interior angles of ABCD is equal to 60 degrees
Solution:
• \(ABCD\) is a quadrilateral.
• \(∠A +∠B +∠C+∠D = 360\)
Statement 1: ABCD is parallelogram.
• By properties of parallelogram
o \(∠A = ∠C=x\)(say), and \(∠B=∠D=y\)(say).
\(2x+2y = 360\)
\(x+y = 180\)
o Here, we don’t know the value of any of the angle.
Hence, statement 1 is not sufficient. We can eliminate answer options A and D.
Statement 2: One of the angles is \(60\)
• With this, we can’t infer anything about the quadrilateral.
Hence, statement 2 is also not sufficient. We can eliminate answer options B.
By combining both the statements.From statement 1: \(x + y = 180\)
From statement 2: One angle is \(60\), either \(x\) is \(60\) or \(y\) is \(60\).
By combining both the statements.
By substituting the value from statement 2 in statement 1 we can find whether any of the angles is \(90\) degrees or not.
Hence, the correct answer is
Option C.Note: - Ideally, we don’t need to do any calculation. By looking at statement 1 we can infer that the opposite angles are equal, and their sum is 360. If we know the value of any of the angles we can find the value of all the angles. In statement 2 value of one angle is given, which means we can find the value of all the angles. Hence the answer is Option C