In cyclic quadrilateral `ABCD,/_A=3/_C` and `/_D=5 /_B`. Find the measure of each angle of the quadrilateral.

Answer

To find the measures of each angle of the cyclic quadrilateral ABCD, where ∠A = 3∠C and ∠D = 5∠B, we can follow these steps:

### Step 1: Set up the equations based on the properties of cyclic quadrilaterals.
In a cyclic quadrilateral, the sum of opposite angles is 180 degrees. Therefore, we can write:
1. ∠A + ∠C = 180° (1)
2. ∠B + ∠D = 180° (2)

### Step 2: Substitute the relationships into the equations.
From the problem, we know:
- ∠A = 3∠C (3)
- ∠D = 5∠B (4)

### Step 3: Substitute equation (3) into equation (1).
Using equation (3) in equation (1):
3∠C + ∠C = 180°
4∠C = 180°
∠C = 180° / 4
∠C = 45°

### Step 4: Find ∠A using the value of ∠C.
Now that we have ∠C, we can find ∠A:
∠A = 3∠C = 3 × 45° = 135°

### Step 5: Substitute equation (4) into equation (2).
Now, we will use equation (4) in equation (2):
∠B + 5∠B = 180°
6∠B = 180°
∠B = 180° / 6
∠B = 30°

### Step 6: Find ∠D using the value of ∠B.
Now that we have ∠B, we can find ∠D:
∠D = 5∠B = 5 × 30° = 150°

### Step 7: Summarize the results.
Now we have all the angles of the cyclic quadrilateral:
- ∠A = 135°
- ∠B = 30°
- ∠C = 45°
- ∠D = 150°

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

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