Three consecutive angles of a cyclic quadrilateral are in the ratio of 1:4: 5. The measure of fourth angle is :

Options

A. `120^@`
B. `60^@`
C. `30^@`
D. `80^@`

Answer

To find the measure of the fourth angle in a cyclic quadrilateral where three consecutive angles are in the ratio of 1:4:5, we can follow these steps:

### Step 1: Define the Angles
Let the three consecutive angles be represented as:
- Angle A = 1x
- Angle B = 4x
- Angle C = 5x

### Step 2: Use the Property of Cyclic Quadrilaterals
In a cyclic quadrilateral, the sum of the opposite angles is equal to 180 degrees. Therefore, we can express the fourth angle (Angle D) in terms of the other angles:
- Angle A + Angle C = 180 degrees
- Angle B + Angle D = 180 degrees

### Step 3: Set Up the Equation
From the first equation:
1x + 5x = 180
This simplifies to:
6x = 180

### Step 4: Solve for x
Now, we can solve for x:
x = 180 / 6
x = 30 degrees

### Step 5: Calculate the Fourth Angle
Now that we have the value of x, we can find the measure of the fourth angle (Angle D) using the second equation:
Angle B + Angle D = 180
Substituting the value of Angle B:
4x + Angle D = 180
4(30) + Angle D = 180
120 + Angle D = 180

### Step 6: Isolate Angle D
Now, isolate Angle D:
Angle D = 180 - 120
Angle D = 60 degrees

### Conclusion
The measure of the fourth angle in the cyclic quadrilateral is 60 degrees.

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