Find the measure of largest angle of a quadrilateral if the measures of its interior angles are in the ratio of `3:4:5:6:`

Options

A. `60^(@)`
B. `120^(@)`
C. `90^(@)`
D. can't be determined

Answer

To find the measure of the largest angle of a quadrilateral with interior angles in the ratio of 3:4:5:6, we can follow these steps:

### Step 1: Define the angles in terms of a variable
Let the angles of the quadrilateral be represented as:
- First angle = \(3x\)
- Second angle = \(4x\)
- Third angle = \(5x\)
- Fourth angle = \(6x\)

### Step 2: Set up the equation for the sum of the angles
We know that the sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore, we can write the equation:
\[ 3x + 4x + 5x + 6x = 360^\circ \]

### Step 3: Combine like terms
Combine the terms on the left side:
\[ (3 + 4 + 5 + 6)x = 360^\circ \]
This simplifies to:
\[ 18x = 360^\circ \]

### Step 4: Solve for \(x\)
To find \(x\), divide both sides of the equation by 18:
\[ x = \frac{360^\circ}{18} = 20^\circ \]

### Step 5: Calculate each angle
Now that we have \(x\), we can find the measures of each angle:
- First angle = \(3x = 3 \times 20^\circ = 60^\circ\)
- Second angle = \(4x = 4 \times 20^\circ = 80^\circ\)
- Third angle = \(5x = 5 \times 20^\circ = 100^\circ\)
- Fourth angle = \(6x = 6 \times 20^\circ = 120^\circ\)

### Step 6: Identify the largest angle
From the calculated angles, we can see:
- \(60^\circ\)
- \(80^\circ\)
- \(100^\circ\)
- \(120^\circ\)

The largest angle is \(120^\circ\).

### Final Answer
The measure of the largest angle of the quadrilateral is \(120^\circ\).

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