If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that `angleC + angleD = k angleAOB`, then the value of k-1 is

Answer

To solve the problem, we need to find the value of \( k - 1 \) given the relationship between the angles in quadrilateral ABCD. Let's break down the solution step by step.

### Step 1: Understand the 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: Express \(\angle C + \angle D\)
From the equation above, we can express \(\angle C + \angle D\) as:
\[ \angle C + \angle D = 360^\circ - (\angle A + \angle B) \]

### Step 3: Use the Angle Bisector Theorem
Since \( O \) is the intersection of the angle bisectors of angles \( A \) and \( B \), we can use the property of angles in triangle \( AOB \):
\[ \angle AOB + \frac{\angle A}{2} + \frac{\angle B}{2} = 180^\circ \]
Multiplying the entire equation by 2 gives:
\[ 2\angle AOB + \angle A + \angle B = 360^\circ \]
From this, we can rearrange to find:
\[ \angle A + \angle B = 360^\circ - 2\angle AOB \]

### Step 4: Substitute into the Expression for \(\angle C + \angle D\)
Now, we substitute \(\angle A + \angle B\) back into our expression for \(\angle C + \angle D\):
\[ \angle C + \angle D = 360^\circ - (360^\circ - 2\angle AOB) \]
This simplifies to:
\[ \angle C + \angle D = 2\angle AOB \]

### Step 5: Relate to Given Equation
According to the problem, we have:
\[ \angle C + \angle D = k \cdot \angle AOB \]
From our previous step, we found that:
\[ \angle C + \angle D = 2\angle AOB \]
Thus, we can equate:
\[ k \cdot \angle AOB = 2\angle AOB \]
This implies:
\[ k = 2 \]

### Step 6: Find \( k - 1 \)
Finally, we need to find \( k - 1 \):
\[ k - 1 = 2 - 1 = 1 \]

### Conclusion
The value of \( k - 1 \) is \( 1 \).

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