Finding Angles of a Cyclic Quadrilateral

Video Explanation

Question

ABCD is a cyclic quadrilateral where: \[ \angle A = (4y + 20)^\circ,\quad \angle B = (3y – 5)^\circ,\quad \angle C = (4x)^\circ,\quad \angle D = (7x + 5)^\circ \] Find all four angles.

Solution

Step 1: Concept

In a cyclic quadrilateral, opposite angles are supplementary:

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

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Step 2: Form Equations

\[ (4y + 20) + 4x = 180 \quad (1) \]

\[ (3y – 5) + (7x + 5) = 180 \quad (2) \]

Simplify: From (1):

\[ 4x + 4y + 20 = 180 \Rightarrow x + y = 40 \quad (3) \]

From (2):

\[ 3y + 7x = 180 \quad (4) \]

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Step 3: Solve Linear Equations

From (3):

\[ y = 40 – x \]

Substitute into (4):

\[ 3(40 – x) + 7x = 180 \]

\[ 120 – 3x + 7x = 180 \]

\[ 4x = 60 \Rightarrow x = 15 \]

Then:

\[ y = 40 – 15 = 25 \]

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Step 4: Find Angles

\[ \angle A = 4y + 20 = 4(25) + 20 = 120^\circ \]

\[ \angle B = 3y – 5 = 75 – 5 = 70^\circ \]

\[ \angle C = 4x = 60^\circ \]

\[ \angle D = 7x + 5 = 105 + 5 = 110^\circ \]

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Conclusion

\[ \angle A = 120^\circ,\quad \angle B = 70^\circ,\quad \angle C = 60^\circ,\quad \angle D = 110^\circ \]

Verification

A + C = \(120 + 60 = 180^\circ\) ✔

B + D = \(70 + 110 = 180^\circ\) ✔