Question

ABC is a triangl BQ and CR are the bisectors of angles \(\angle\)ABC and \(\angle\)BCA respectively, Q and R are two points on the sides AC and AB respectively. The bisectors meet at a point O. If AQOR is a cyclic quadrilateral, find the measure of \(\angle\)BAC

• A. 90°
• B. 45°
• C. 30°
• D. 60°
• E. None of these

Answer 1

The Correct Option is A

Step 1: Understand the given information.
- \( \triangle ABC \) is a triangle.
- \( BQ \) and \( CR \) are the angle bisectors of \( \angle ABC \) and \( \angle BCA \) respectively.
- Points \( Q \) and \( R \) lie on sides \( AC \) and \( AB \) respectively.
- The angle bisectors meet at a point \( O \).
- \( AQOR \) is a cyclic quadrilateral.
We are asked to find the measure of \( \angle BAC \).

Step 2: Use the property of cyclic quadrilaterals.
In a cyclic quadrilateral, opposite angles sum up to 180°. Hence, we can use this property to find the required angle.

Step 3: Apply the cyclic quadrilateral property.
Since \( AQOR \) is a cyclic quadrilateral, we have the following relation:
\( \angle AQR + \angle AOR = 180^\circ \).
\( \angle BAC + \angle AQR = 180^\circ \) (since \( \angle AQR \) and \( \angle BAC \) are opposite angles in the cyclic quadrilateral).

Step 4: Conclusion.
Using the properties of cyclic quadrilaterals and angle bisectors, the measure of \( \angle BAC \) is \( 90^\circ \).

Final Answer:
The correct option is (A): 90°.