Find the angles of a parallelogram if one angle is three times another.

Options

A. 45°, 135°, 45°, 135°
B. 50°, 130°, 50°, 130°
C. 40°, 140°, 40°, 140°
D. 55°, 125°, 55°, 125°

Answer

To find the angles of a parallelogram where one angle is three times another, we can follow these steps:

### Step 1: Understand the properties of a parallelogram
In a parallelogram, opposite angles are equal, and the sum of the consecutive interior angles is 180 degrees.

### Step 2: Define the angles
Let’s denote one angle as \( \alpha \) and the other angle as \( \beta \). According to the problem, we know that:
\[ \alpha = 3\beta \]

### Step 3: Use the property of consecutive angles
Since \( \alpha \) and \( \beta \) are consecutive angles in the parallelogram, we can use the property that their sum is 180 degrees:
\[ \alpha + \beta = 180^\circ \]

### Step 4: Substitute the value of \( \alpha \)
Now, substitute \( \alpha \) in the equation:
\[ 3\beta + \beta = 180^\circ \]
This simplifies to:
\[ 4\beta = 180^\circ \]

### Step 5: Solve for \( \beta \)
Now, divide both sides by 4 to find \( \beta \):
\[ \beta = \frac{180^\circ}{4} = 45^\circ \]

### Step 6: Find \( \alpha \)
Now that we have \( \beta \), we can find \( \alpha \):
\[ \alpha = 3\beta = 3 \times 45^\circ = 135^\circ \]

### Step 7: List all angles
Since opposite angles are equal in a parallelogram, the angles are:
- \( \alpha = 135^\circ \)
- \( \beta = 45^\circ \)
- Therefore, the angles of the parallelogram are:
- \( 135^\circ, 45^\circ, 135^\circ, 45^\circ \)

### Final Answer
The angles of the parallelogram are \( 135^\circ, 45^\circ, 135^\circ, 45^\circ \).

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