The angles of a quadrilateral are in the ratio `3:5:7:9`. Find the measure of each of these angles.

Options

A. `45^(@), 75^(@), 105^(@), 165^(@)`
B. `35^(@), 75^(@), 105^(@), 135^(@)`
C. `45^(@), 75^(@), 105^(@), 135^(@)`
D. `45^(@), 85^(@), 105^(@), 135^(@)`

Answer

To find the measures of the angles of a quadrilateral given in the ratio \(3:5:7:9\), we can follow these steps:

### Step 1: Set Up the Angles
Let the angles of the quadrilateral be represented as follows:
- Angle A = \(3x\)
- Angle B = \(5x\)
- Angle C = \(7x\)
- Angle D = \(9x\)

### Step 2: Use the Sum of Angles in a Quadrilateral
We know that the sum of the angles in any quadrilateral is \(360^\circ\). Therefore, we can write the equation:
\[ 3x + 5x + 7x + 9x = 360^\circ \]

### Step 3: Combine Like Terms
Combine the terms on the left side:
\[ (3 + 5 + 7 + 9)x = 360^\circ \]
\[ 24x = 360^\circ \]

### Step 4: Solve for \(x\)
To find the value of \(x\), divide both sides by 24:
\[ x = \frac{360^\circ}{24} = 15^\circ \]

### Step 5: Calculate Each Angle
Now that we have the value of \(x\), we can find each angle:
- Angle A = \(3x = 3 \times 15^\circ = 45^\circ\)
- Angle B = \(5x = 5 \times 15^\circ = 75^\circ\)
- Angle C = \(7x = 7 \times 15^\circ = 105^\circ\)
- Angle D = \(9x = 9 \times 15^\circ = 135^\circ\)

### Step 6: Write the Final Answers
The measures of the angles in the quadrilateral are:
- Angle A = \(45^\circ\)
- Angle B = \(75^\circ\)
- Angle C = \(105^\circ\)
- Angle D = \(135^\circ\)