1. An angle in a segment smaller than a semicircle is an ____.
2. The angle subtended by a segment of a circle greater than a semicircle is _____.
3. An angle subtended by a segment smaller than a semicircle is an obtuse angle.
4. The angle subtended by a segment smaller than a semicircle is — —.
5. Prove that an angle in a segment greater than a semicircle is an acute angle.
6. The angle in a circular segment larger than a semicircle is an obtuse angle.
7. An angle in a segment greater than a semicircle is an obtuse angle.
8. An angle in a segment larger than a semicircle is an obtuse angle.
9. Prove with reasoning that an angle in a segment greater than a semicircle is an acute angle.
10. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.
11. The measure of all angles in a circular segment is ______.
12. An angle in a segment larger than a semicircle.
(a) Acute angle. (b) Obtuse angle. (c) Right angle. (d) Straight angle.
13. Angle in a semicircle is a _____.
14. Points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\) of triangle \(∆PQR\), respectively. Given: \(PQ = 8\) units, \(YR = 12\) units, \(PY = 4\) units, and the length of \(PY\) is 2 units less than the length of \(XQ\). Determine, with reasoning, whether segment \(XY\) is parallel to side \(QR\).
15. Angle in a segment greater than a semicircle.
(a) right angle. (b) acute angle. (c) straight angle (d) obtuse angle.
16. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.
17. "Angle in a semicircle is a right angle — prove it."
18. In a semicircle with a radius of 4 cm, AB is the diameter and ∠ACB is an angle inscribed in the semicircle. If BC = \(2\sqrt{7}\) cm, find the length of AC.
19. Prove that the line segment joining the midpoints of two sides of a triangle is equal to half of the third side.
20. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.
21. The base of a triangle is 18 meters more than twice its height. If the area of the triangle is 360 square meters, find its height.
22. Prove that an angle in a semicircle is a right angle.
23. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.
24. Draw a median \(AD\) of \(\triangle ABC\). If a straight line parallel to \(BC\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that \(AD\) bisects the line segment \(PQ\).
25. If the ratio of the lengths of two corresponding sides of two similar triangles is 7:11, then their perimeter ratio is _____.
26. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.
(a) 30° (b) 45° (c) 60° (d) 90°
27. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____.
28. The circular measure of each interior angle of a regular hexagon is _____.
(a) \(\cfrac{\pi^c}{4}\) (b) \(\cfrac{\pi^c}{6}\) (c) \(\cfrac{\pi^c}{3}\) (d) \(\cfrac{2\pi^c}{3}\)
29. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters. The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\). Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters. From \( \triangle \)ABO, we get: \(\cfrac{AB}{BO} = \tan \theta\) Or, \(\cfrac{x}{60} = \tan\theta\) --------(i) From \( \triangle \)COD, we get: \(\cfrac{CD}{OD} = \tan(90^o - \theta)\) Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii) Multiplying equations (i) and (ii), we get: \(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\) Or, \(\cfrac{2x^2}{60 \times 60} = 1\) Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\) Or, \(x = 30\sqrt2\) \(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters. And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters. (Proved).
30. The measure of an angle subtended by a semicircle is _____.
