1. If the opposite angles of a quadrilateral are supplementary, then its vertices lie on the same circle.
2. If the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on a circle.
3. Prove that if the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral lie on a circle (i.e., the quadrilateral is cyclic).
4. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.
5. If the corresponding angles of two quadrilaterals are equal, then the two quadrilaterals are similar.
6. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?
7. If the three consecutive angles of a cyclic quadrilateral are in the ratio \(1:2:5\), then the measure of the fourth angle will be _____.
8. If three consecutive angles of a cyclic quadrilateral are in the ratio \(1:3:4\), then the measure of the fourth angle will be _____.
9. If the corresponding angles of two quadrilaterals are equal, then the two quadrilaterals are similar.
10. If the corresponding angles of two quadrilaterals are equal, then the quadrilaterals are similar.
11. If two triangles have their bases on the same straight line and share the same vertex (the opposite vertex), then the ratio of their areas is equal to the ratio of the lengths of their bases.
12. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.
13. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?
(a) 14 (b) 44 (c) 20 (d) 24
14. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?
(a) 145° (b) 55° (c) 110° (d) None of the above
15. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.
16. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
17. If three angles of a quadrilateral are \(\frac{π}{3}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the measure of the fourth angle in both sexagesimal (degree) and circular (radian) units.
18. If the angles of depression from a lighthouse to two ships located along the same straight line are 60° and 30°, and the nearer ship is 150 meters away from the lighthouse, then what is the distance of the farther ship from the lighthouse?
19. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
20. If the ratio of three consecutive angles of a cyclic quadrilateral is 1 : 2 : 3, what are the measures of the first and third angles?
21. If three angles of a quadrilateral are \(\frac{π}{5}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the sexagesimal (degree) and circular (radian) measure of the fourth angle.
22. From the roof and the base of a 16-meter high building, the angles of elevation to the top of a temple are 45° and 60°, respectively. Find the height of the temple and its horizontal distance from the building. (The base of the building and the temple lie on the same horizontal plane.)
23. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.
24. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
25. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
26. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
27. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
28. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
29. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.
30. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]
