In a trapezium ABCD where AB || DC, diagonals AC and BD intersect at O. If ∠BAC = 45°, then what is the value of ∠OCD?

If B is the center of a circle and lines AD and CD are tangent segments drawn from an external point D, then which of the following classifications apply to quadrilateral ABCD? (a) Kite (b) Cyclic quadrilateral (c) Rhombus (d) Trapezium

In a parallelogram PQRS, rays PA and SA are the angle bisectors of ∠QPS and ∠RSP respectively. Find the measure of ∠PAS.

In ΔPQR, points A, B, and C are the midpoints of sides PQ, QR, and PR respectively. Points L, M, and N are the midpoints of sides AC, AB, and BC respectively. If the lengths of the sides of ΔPQR are 15.5 cm, 12.7 cm, and 9.4 cm, find the perimeter of ΔLMN.

In which type of triangle does the circumcenter (intersection point of perpendicular bisectors of the sides) lie outside the triangle?

If the angles of a quadrilateral LMNP are in the ratio 4 : 5 : 7 : 8, what type of quadrilateral is LMNP?

From the given sets of side lengths, which ones can form a valid triangle? (a) 2.1 cm, 4 cm, 1.9 cm (b) 5.7 cm, 3.2 cm, 2.4 cm (c) 1.1 cm, 0.7 cm, 0.5 cm (d) 3.6 cm, 4.8 cm, 6 cm

Find the radius of the circle in which a central angle of 30° intercepts an arc of length 32 cm. (Use π ≈ 3.1416)

What is the primary meaning of the term 'gradient' in mathematics?