In a triangle `ABC, /_A = 70^@, /_B = 80^@` and D is the incentre of `DeltaABC. /_ACB = 2 xx` and `/_BDC = y^@`. The values of x and y, respectively are.

Options

A. 15, 130
B. 15, 125
C. 35, 40
D. 30, 150

Answer

To solve the problem step by step, we will analyze the triangle ABC with the given angles and find the values of x and y.

### Step 1: Find Angle C
Given:
- Angle A = 70°
- Angle B = 80°

Using the property that the sum of angles in a triangle is 180°, we can find angle C:

\[ \text{Angle C} = 180° - (\text{Angle A} + \text{Angle B}) = 180° - (70° + 80°) = 180° - 150° = 30° \]

### Step 2: Determine the Incenter and Angle ACB
Since D is the incenter of triangle ABC, it bisects angle C. Therefore, angle ACB can be expressed as:

\[ \text{Angle ACB} = 2x \]

From Step 1, we found that angle C = 30°. Thus:

\[ 2x = 30° \]

To find x, we divide both sides by 2:

\[ x = \frac{30°}{2} = 15° \]

### Step 3: Find Angle BDC
Now, we need to find angle BDC (denoted as y). In triangle BDC, we know:
- Angle DBC = 40° (since angle B = 80° and D bisects it, angle DBC = 80°/2 = 40°)
- Angle DCB = 15° (since angle ACB = 2x = 30°, and D bisects it, angle DCB = 30°/2 = 15°)

Now, we can find angle BDC using the triangle angle sum property:

\[ \text{Angle BDC} + \text{Angle DBC} + \text{Angle DCB} = 180° \]

Substituting the known values:

\[ y + 40° + 15° = 180° \]

Combining the angles:

\[ y + 55° = 180° \]

Now, solving for y:

\[ y = 180° - 55° = 125° \]

### Final Values
Thus, we have:
- \( x = 15° \)
- \( y = 125° \)

### Summary
The values of x and y are:
- \( x = 15° \)
- \( y = 125° \)