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A walking route in Darwin forms a quadrilateral ABCD where AB is parallel to DC, AD is parallel to BC, and diagonal AC bisects angle A. If angle A is 90 degrees, what is the exact measure, in degrees, of angle BAC?
Use the properties of parallelograms and angle bisectors to determine the answer.
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Worked solution
- Given quadrilateral ABCD in Darwin with AB parallel to DC and AD parallel to BC, so ABCD is a parallelogram by definition of opposite sides being parallel.
- Angle A is given as 90 degrees, so ∠DAB = 90°.
- Diagonal AC bisects angle A, so ∠BAC = ∠CAD. Since ∠DAB = 90°, and AC bisects it, each half is 90° / 2 = 45°.
- Therefore, ∠BAC = 45°.
- This result is consistent with the properties of a parallelogram: opposite angles are equal, adjacent angles sum to 180°, and diagonals bisect angles only in special cases like rectangles and rhombuses.
- Since angle A is 90° and ABCD is a parallelogram, and one diagonal bisects a right angle, the shape must be a rectangle (all angles 90°) or a square. But only in a square do diagonals bisect angles into 45°, and here only angle A is specified as 90°, so it could be a rectangle with diagonal bisecting angle A.
- In a rectangle, diagonals do not generally bisect angles unless it is a square. But here, diagonal AC bisects angle A, which is 90°, so each half is 45°. This only occurs if the adjacent sides are equal, meaning AB = AD.
- So, since AB = AD and angle A is 90°, ABCD is a square. In a square, diagonals bisect the 90° angles into two 45° angles.
- Therefore, angle BAC, which is half of angle A, is 45°.
- Check: in a square, diagonal AC creates two 45°-45°-90° triangles. So ∠BAC = 45° is correct.
- Final verification: since ABCD is a square (as deduced from AB || DC, AD || BC, angle A = 90°, and diagonal bisecting angle A), then ∠BAC = 45°.
- The exact measure of angle BAC is 45 degrees.
Reference answer: 45
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