In a parallelogram ABCD if `angleA=(3x-20)^(@), angleB=(y+15)^(@) and angleC=(x+40)^(@)`, then the values of x and y respectgively are

Options

A. `30^(@),95^(@)`
B. `95^(@), 30^(@)`
C. `60^(@),30^(@)`
D. `30^(@),60^(@)`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) in the given parallelogram \( ABCD \) where:

- \( \angle A = (3x - 20)^\circ \)
- \( \angle B = (y + 15)^\circ \)
- \( \angle C = (x + 40)^\circ \)

### Step 1: Use the property of opposite angles in a parallelogram
In a parallelogram, opposite angles are equal. Therefore, we can set up the following equation:

\[ \angle A = \angle C \]

Substituting the expressions for angles A and C:

\[ 3x - 20 = x + 40 \]

### Step 2: Solve for \( x \)
Now, we will solve the equation for \( x \):

1. Rearranging the equation gives:
\[ 3x - x = 40 + 20 \]
\[ 2x = 60 \]

2. Dividing both sides by 2:
\[ x = 30 \]

### Step 3: Find \( \angle A \) and \( \angle C \)
Now that we have \( x \), we can find the measures of angles A and C:

1. Substitute \( x \) back into the expression for \( \angle A \):
\[ \angle A = 3(30) - 20 = 90 - 20 = 70^\circ \]

2. Substitute \( x \) back into the expression for \( \angle C \):
\[ \angle C = 30 + 40 = 70^\circ \]

### Step 4: Use the property of adjacent angles in a parallelogram
The sum of adjacent angles in a parallelogram is 180 degrees. Therefore, we can write:

\[ \angle A + \angle B = 180^\circ \]

Substituting the known value of \( \angle A \) and the expression for \( \angle B \):

\[ 70 + (y + 15) = 180 \]

### Step 5: Solve for \( y \)
Now, we will solve the equation for \( y \):

1. Simplifying the left side:
\[ 70 + y + 15 = 180 \]
\[ y + 85 = 180 \]

2. Subtracting 85 from both sides:
\[ y = 180 - 85 = 95 \]

### Final Answer
The values of \( x \) and \( y \) are:

\[ x = 30, \quad y = 95 \]