The angles of a triangle are in the ratio of 4 : 4 : 2. What is the measure of the smallest angle (in degrees)?

Options

A. 72
B. 36
C. 44
D. 54

Answer

To find the measure of the smallest angle in a triangle where the angles are in the ratio of 4:4:2, we can follow these steps:

### Step 1: Understand the Ratio
The angles of the triangle are in the ratio of 4:4:2. This means we can express the angles in terms of a variable. Let the common multiple be \( x \). Therefore, the angles can be expressed as:
- Angle 1 = \( 4x \)
- Angle 2 = \( 4x \)
- Angle 3 = \( 2x \)

### Step 2: Set Up the Equation
We know that the sum of the angles in a triangle is 180 degrees. Therefore, we can set up the equation:
\[ 4x + 4x + 2x = 180 \]

### Step 3: Simplify the Equation
Combine the terms on the left side:
\[ 10x = 180 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 10 to find \( x \):
\[ x = \frac{180}{10} = 18 \]

### Step 5: Calculate the Angles
Now that we have \( x \), we can find the measures of the angles:
- Angle 1 = \( 4x = 4 \times 18 = 72 \) degrees
- Angle 2 = \( 4x = 4 \times 18 = 72 \) degrees
- Angle 3 = \( 2x = 2 \times 18 = 36 \) degrees

### Step 6: Identify the Smallest Angle
Among the angles calculated, the smallest angle is:
\[ 36 \text{ degrees} \]

### Final Answer
The measure of the smallest angle is **36 degrees**.

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