In a triangle `ABC,angleA=70^(@),angleB=80^(@)` and D is the incentre of `DeltaABC,angleACB=2x^(@)` and `angleBDC=y^(@)`. The values of x and y respectively will be

Options

A. `15^(@),130^(@)`
B. `15^(@),125^(@)`
C. `35^(@),40^(@)`
D. `30^(@),150^(@)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find Angle C
In triangle ABC, the sum of the angles is 180 degrees. We know:
- Angle A = 70 degrees
- Angle B = 80 degrees
- Angle C = 2x degrees

Using the angle sum property:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Substituting the known values:
\[ 70^\circ + 80^\circ + 2x = 180^\circ \]
This simplifies to:
\[ 150^\circ + 2x = 180^\circ \]

### Step 2: Solve for x
Now, we can isolate \(2x\):
\[ 2x = 180^\circ - 150^\circ \]
\[ 2x = 30^\circ \]
Now, divide by 2 to find \(x\):
\[ x = \frac{30^\circ}{2} = 15^\circ \]

### Step 3: Find Angle BDC
Since D is the incenter of triangle ABC, we can use the property of the incenter to find angle BDC (y):
\[ \text{Angle BDC} = 90^\circ + \frac{\text{Angle A}}{2} \]
Substituting the value of Angle A:
\[ y = 90^\circ + \frac{70^\circ}{2} \]
Calculating \(\frac{70^\circ}{2}\):
\[ y = 90^\circ + 35^\circ \]
\[ y = 125^\circ \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = 15^\circ \quad \text{and} \quad y = 125^\circ \]