Углы треугольника лежат в плоскости AP, и наибольший угол вдвое больше наименьшего. Найдите все у...

To solve this problem, we will use the properties of arithmetic progression (AP) and the fact that the sum of angles in a triangle is 180°.Finding the angles in degreesDefine the angles: Let the three angles of the triangle be \(a-d\), \(a\), and \(a+d\), where \(a\) is the middle angle and \(d\) is the common difference. For the angles to be in increasing order, we can assume \(d0\).Use the angle sum property: The sum of the angles in any triangle is 180°.\((a-d)+a+(a+d)=180^{\circ }\)\(3a=180^{\circ }\)\(a=60^{\circ }\).So, the middle angle is 60°.Use the greatest and least angle relationship: The problem states that the greatest angle is double the least angle.Greatest angle = \(a+d\)Least angle = \(a-d\)Therefore, \(a+d=2(a-d)\).Solve for the common difference (\(d\)):\(a+d=2a-2d\)\(3d=a\)Substitute the value of \(a\) from Step 2:\(3d=60^{\circ }\)\(d=20^{\circ }\)Calculate all the angles:Least angle: \(a-d=60^{\circ }-20^{\circ }=40^{\circ }\).Middle angle: \(a=60^{\circ }\).Greatest angle: \(a+d=60^{\circ }+20^{\circ }=80^{\circ }\).Converting the angles to radiansTo convert from degrees to radians, use the formula: radians = degrees \(\times \) \(\frac{\pi }{180}\).Least angle (40°):\(40\times \frac{\pi }{180}=\frac{4\pi }{18}=\frac{2\pi }{9}\) radians.Middle angle (60°):\(60\times \frac{\pi }{180}=\frac{6\pi }{18}=\frac{\pi }{3}\) radians.Greatest angle (80°):\(80\times \frac{\pi }{180}=\frac{8\pi }{18}=\frac{4\pi }{9}\) radians. 16(2π/5) m = 88 X (88X222) m = 70 m. 72° P Hence, the length of the rope is 70 m. rm A 1. Using a protractor, draw each of the following angles. (i) 60° EXERCISE 14 2. (ⅰ)( (vii) - (v) -40° (ii) 130° (iii) 300° (vi) -220° (vii) -310° (viii) -400° 2. Express each of the following angles in radians. (i) 36° (ii) 120° (v) 400° (vi) 7°30′ (iii) 225° (iv) 330° (vii) -270° 3. Express each of the following angles in degrees. (iii) 18π 5 (iv)-4 4. The angles of a triangle are in AP and the greatest angle is double the least. Find all the angles in degrees and radians. 5. The difference between the two acute angles of a right triangle is (4) Find these angles in radians and degrees. 6. Find the radius of a circle in which a central angle of 45° intercepts an arc of length 33 cm. (Take π = 22/7.) (iv) 430° 3. (i) 75 4. (40° 6. 42 ст ()() (viii) -(22° 30′) 11. (11 49 4. Let t Then Also, 7. Find the length of an arc of a circle of radius 14 cm which subtends an angle of 36° at the centre.