The measures of three angles of a quadrilateral are in the ratio 1: 2:3. If the sum of these three measures is equal to the measure of the fourth angle, find the smallest angle.

Options

A. `30^@`
B. `40^@`
C. `60^@`
D. `50^@`

Answer

To solve the problem step by step, we will follow the given information and use algebra to find the smallest angle of the quadrilateral.

### Step 1: Define the angles
Let the measures of the three angles of the quadrilateral be represented in terms of a variable \( x \):
- First angle = \( 1x \)
- Second angle = \( 2x \)
- Third angle = \( 3x \)

### Step 2: Express the fourth angle
According to the problem, the sum of the three angles is equal to the measure of the fourth angle. Therefore, we can express the fourth angle \( y \) as:
\[ y = 1x + 2x + 3x \]
\[ y = 6x \]

### Step 3: Use the property of quadrilaterals
The sum of all angles in a quadrilateral is always \( 360^\circ \). Therefore, we can write the equation:
\[ 1x + 2x + 3x + y = 360^\circ \]
Substituting \( y \) with \( 6x \):
\[ 1x + 2x + 3x + 6x = 360^\circ \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ (1x + 2x + 3x + 6x) = 12x \]
Thus, we have:
\[ 12x = 360^\circ \]

### Step 5: Solve for \( x \)
Now, divide both sides by 12 to find \( x \):
\[ x = \frac{360^\circ}{12} \]
\[ x = 30^\circ \]

### Step 6: Find the angles
Now that we have \( x \), we can find the measures of the three angles:
- First angle = \( 1x = 1 \times 30^\circ = 30^\circ \)
- Second angle = \( 2x = 2 \times 30^\circ = 60^\circ \)
- Third angle = \( 3x = 3 \times 30^\circ = 90^\circ \)

### Step 7: Find the fourth angle
Now, we can find the fourth angle:
\[ y = 6x = 6 \times 30^\circ = 180^\circ \]

### Step 8: Identify the smallest angle
The smallest angle among \( 30^\circ, 60^\circ, 90^\circ, \) and \( 180^\circ \) is:
\[ \text{Smallest angle} = 30^\circ \]

### Final Answer
The smallest angle is \( 30^\circ \).

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