Finding Angles of a Cyclic Quadrilateral

Video Explanation

Question

In a cyclic quadrilateral ABCD: \[ \angle A = (2x + 4)^\circ,\quad \angle B = (y + 3)^\circ,\quad \angle C = (2y + 10)^\circ,\quad \angle D = (4x – 5)^\circ \] Find all four angles.

Solution

Step 1: Concept

Opposite angles of a cyclic quadrilateral are supplementary:

\[ \angle A + \angle C = 180^\circ,\quad \angle B + \angle D = 180^\circ \]

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Step 2: Form Equations

\[ (2x + 4) + (2y + 10) = 180 \]

\[ 2x + 2y + 14 = 180 \Rightarrow x + y = 83 \quad (1) \]

\[ (y + 3) + (4x – 5) = 180 \]

\[ 4x + y – 2 = 180 \Rightarrow 4x + y = 182 \quad (2) \]

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Step 3: Solve Linear Equations

From (1):

\[ y = 83 – x \]

Substitute into (2):

\[ 4x + (83 – x) = 182 \]

\[ 3x + 83 = 182 \]

\[ 3x = 99 \Rightarrow x = 33 \]

Then:

\[ y = 83 – 33 = 50 \]

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Step 4: Find Angles

\[ \angle A = 2x + 4 = 66 + 4 = 70^\circ \]

\[ \angle B = y + 3 = 50 + 3 = 53^\circ \]

\[ \angle C = 2y + 10 = 100 + 10 = 110^\circ \]

\[ \angle D = 4x – 5 = 132 – 5 = 127^\circ \]

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Conclusion

\[ \angle A = 70^\circ,\quad \angle B = 53^\circ,\quad \angle C = 110^\circ,\quad \angle D = 127^\circ \]

Verification

A + C = \(70 + 110 = 180^\circ\) ✔

B + D = \(53 + 127 = 180^\circ\) ✔