In a cyclic quadrilateral ABCD ` angle A : angle C = 3: 1 and ` angle B : angle D = 1 : 5 ` find each angle of the quadrilateral

Answer

To solve the problem of finding the angles of the cyclic quadrilateral ABCD given the ratios of the angles, we can follow these steps:

### Step 1: Define the Angles
Given the ratios:
- Angle A : Angle C = 3 : 1
- Angle B : Angle D = 1 : 5

Let:
- Angle A = 3x
- Angle C = x

And for angles B and D:
- Angle B = y
- Angle D = 5y

### Step 2: Use the Property of Cyclic Quadrilaterals
In a cyclic quadrilateral, the sum of the opposite angles is 180 degrees. Therefore, we can write two equations based on this property:
1. Angle A + Angle C = 180°
2. Angle B + Angle D = 180°

### Step 3: Substitute the Values
From the first equation:
\[ 3x + x = 180° \]
This simplifies to:
\[ 4x = 180° \]
Now, solve for x:
\[ x = \frac{180°}{4} = 45° \]

### Step 4: Find Angles A and C
Now that we have x, we can find the angles:
- Angle C = x = 45°
- Angle A = 3x = 3 * 45° = 135°

### Step 5: Use the Second Equation
Now, using the second equation:
\[ y + 5y = 180° \]
This simplifies to:
\[ 6y = 180° \]
Now, solve for y:
\[ y = \frac{180°}{6} = 30° \]

### Step 6: Find Angles B and D
Now that we have y, we can find the angles:
- Angle B = y = 30°
- Angle D = 5y = 5 * 30° = 150°

### Step 7: Summary of the Angles
Now, we can summarize the angles:
- Angle A = 135°
- Angle B = 30°
- Angle C = 45°
- Angle D = 150°

### Final Answer
The angles of the cyclic quadrilateral ABCD are:
- Angle A = 135°
- Angle B = 30°
- Angle C = 45°
- Angle D = 150°