In a cyclic quadrilateral ABCD, \(\angle A = (4x + 3)^\circ\), \(\angle B = (3y + 9)^\circ\), \(\angle C = (4y - 3)^\circ\), \(\angle D = (5x - 10)^\circ\). What is \(\angle A + \angle B\) equal to?
\(170^\circ\)
In a cyclic quadrilateral, opposite angles sum to \(180^\circ\). So \(\angle A + \angle C = 180^\circ \Rightarrow (4x+3)+(4y-3) = 180 \Rightarrow 4x+4y = 180 \Rightarrow x+y = 45\). Also \(\angle B + \angle D = 180^\circ \Rightarrow (3y+9)+(5x-10) = 180 \Rightarrow 5x+3y = 181\). Substituting \(y = 45-x\): \(5x + 3(45-x) = 181 \Rightarrow 2x = 46 \Rightarrow x = 23\), so \(y = 22\). Then \(\angle A = 4(23)+3 = 95^\circ\), \(\angle B = 3(22)+9 = 75^\circ\). So \(\angle A + \angle B = 170^\circ\).
Similar Questions
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: ABCD is a rhombus of side 10 cm. What is the sum of the squares of the diagonals of the rhombus?
Statement I: \(AC:BD = 3:4\).
Statement II: \(AC = 6\) cm.
Which one of the following is correct in respect of the above Question and Statements?
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