The angles of quadrilateral are `x-20^(@), 2x^(@), x+20^(@)`, and `x+30^(@)`. Find the smallest angle of the quadrilateral.

Options

A. `66^@`
B. `56^@`
C. `36^@`
D. `46^@`

Answer

To find the smallest angle of the quadrilateral with angles given as \( x - 20^\circ \), \( 2x \), \( x + 20^\circ \), and \( x + 30^\circ \), we can follow these steps:

### Step 1: Write the equation for the sum of the angles of a quadrilateral.
The sum of the interior angles of a quadrilateral is always \( 360^\circ \). Therefore, we can set up the equation:

\[ (x - 20) + (2x) + (x + 20) + (x + 30) = 360 \]

### Step 2: Combine like terms.
Now, let's combine the \( x \) terms and the constant terms:

\[ x - 20 + 2x + x + 20 + x + 30 = 360 \]

Combining the \( x \) terms:

\[ x + 2x + x + x = 5x \]

Now combining the constants:

\[ -20 + 20 + 30 = 30 \]

So, we can rewrite the equation as:

\[ 5x + 30 = 360 \]

### Step 3: Solve for \( x \).
Next, we will isolate \( x \) by subtracting \( 30 \) from both sides:

\[ 5x = 360 - 30 \]
\[ 5x = 330 \]

Now, divide both sides by \( 5 \):

\[ x = \frac{330}{5} = 66 \]

### Step 4: Find the angles.
Now that we have the value of \( x \), we can find each angle:

1. \( x - 20 = 66 - 20 = 46^\circ \)
2. \( 2x = 2 \times 66 = 132^\circ \)
3. \( x + 20 = 66 + 20 = 86^\circ \)
4. \( x + 30 = 66 + 30 = 96^\circ \)

### Step 5: Identify the smallest angle.
Now, we compare the angles:

- \( 46^\circ \)
- \( 132^\circ \)
- \( 86^\circ \)
- \( 96^\circ \)

The smallest angle is \( 46^\circ \).

### Final Answer:
The smallest angle of the quadrilateral is \( 46^\circ \).

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