1. The first question asks for the sum of angles \(\angle BAD + \angle BCD\) in a quadrilateral. 2. Given the options: ① 90° ② 270° ③ 180° ④ 360°, the sum of opposite angles in a cyclic quadrilateral is 180°. 3. For the statements about quadrilateral ABCD: i. If ABC is a pentagon, \(\angle ADC\) is acute (sushmokon). ii. If \(\angle BAD = 45^\circ\) and \(\angle BCD = 45^\circ\), then their sum is 90°. iii. \(\angle ABC + \angle ADC = 180^\circ\) is true for cyclic quadrilaterals. 4. The true statements are i and iii, so option ②. 5. For the circle with center O and tangents from point P touching at A and B: - Tangents from a point to a circle are equal, so \(PA = PB\). - \(\angle OPA = \angle OAP\) because of tangent properties. 6. For the trigonometric identity \(\sin^2 45^\circ + \cos^2 A = 1\), since \(\sin^2 45^\circ = \frac{1}{2}\), then \(\cos^2 A = \frac{1}{2}\), so \(A = 45^\circ\). 7. For \(\sin(90^\circ - \theta) = \cos \theta\) and \(\cot(90^\circ - \theta) = \tan \theta\), the correct pair is option ②. 8. For the sector of a circle with radius 4 cm and central angle 60°: - Area of sector = \(\frac{\theta}{360} \times \pi r^2 = \frac{60}{360} \times \pi \times 4^2 = \frac{1}{6} \times \pi \times 16 = \frac{16\pi}{6} = \frac{8\pi}{3} \approx 8.37\) cm². 9. For the data set 39, 40, 44, 45, 45, 50: - Mean = 43.833 - Median = 44.5 - Mode = 45 10. The true statements are i and ii. Final answers: - Sum of \(\angle BAD + \angle BCD = 90^\circ\) (option ①) - True statements about ABCD: ② (i and iii) - Tangents equality: \(PA = PB\) (option ②) - \(\angle OPA = \angle OAP\) (option ①) - \(A = 45^\circ\) (option ③) - \(\sin(90^\circ - \theta) = \cos \theta\), \(\cot(90^\circ - \theta) = \tan \theta\) (option ②) - Area of sector = 8.37 cm² (option ③) - True data statements: i and ii