In a quadrilateral `ABCD, angle D=60^(@) and angle C = 100^(@)`. The bisectors of `angle A and angle B` meet at the point P. The measure of `angle APB` is

Options

A. `80^(@)`
B. `70^(@)`
C. `100^(@)`
D. `60^(@)`

Answer

To find the measure of angle \( APB \) in quadrilateral \( ABCD \) where \( \angle D = 60^\circ \) and \( \angle C = 100^\circ \), we can follow these steps:

### Step 1: Determine the sum of angles in quadrilateral \( ABCD \)
In any quadrilateral, the sum of the interior angles is \( 360^\circ \). Therefore, we can write:

\[ \angle A + \angle B + \angle C + \angle D = 360^\circ \]

### Step 2: Substitute the known angles
Substituting the known values of \( \angle C \) and \( \angle D \):

\[ \angle A + \angle B + 100^\circ + 60^\circ = 360^\circ \]

### Step 3: Simplify the equation
Now, simplify the equation:

\[ \angle A + \angle B + 160^\circ = 360^\circ \]

Subtract \( 160^\circ \) from both sides:

\[ \angle A + \angle B = 360^\circ - 160^\circ = 200^\circ \]

### Step 4: Find the measures of angles \( A \) and \( B \)
Since \( \angle A + \angle B = 200^\circ \), we can denote:

\[ \angle A + \angle B = 200^\circ \]

### Step 5: Use the angle bisector theorem
Let \( \angle PAB = x \) and \( \angle ABP = y \). Since \( P \) is the intersection of the angle bisectors of \( A \) and \( B \):

\[ x + y = 100^\circ \]

### Step 6: Apply the triangle angle sum property
In triangle \( APB \):

\[ \angle APB + x + y = 180^\circ \]

### Step 7: Substitute the known values
Substituting \( x + y = 100^\circ \) into the triangle angle sum equation:

\[ \angle APB + 100^\circ = 180^\circ \]

### Step 8: Solve for \( \angle APB \)
Now, isolate \( \angle APB \):

\[ \angle APB = 180^\circ - 100^\circ = 80^\circ \]

Thus, the measure of angle \( APB \) is \( 80^\circ \).

### Final Answer
\[ \angle APB = 80^\circ \]

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