From the diagram below, if side DF is 22 cm., side CB would be  ______.

AD is a straight lineAngle CBD is 162ºWhat is the size of angle ACB?I DON'T KNOWSUBMIT ANSWER

A triangle DEF has DF=13 cm, angle EDF is equal to 37 degrees and angle DFE is equal to 68 degrees. The side length DE (to the nearest cm) is equal to

Two sides of a parallelogram are 21 cm and 14 cm. If the height corresponding to the side of length 21 cm is 6 cm long, the length of the height corresponding to the side 14 cm will be ______.

In ΔABC, AC = 15 cm and DE || BC. If ADDB=21, then EC = ______.

To solve this problem, we need to use properties of circles and angles. Let's go through each part step by step. ### Given: - \( \angle CBD = 42^\circ \) - \( \angle OBE = 20^\circ \) - \( BC \) is a tangent to the circle at \( B \) ### To find: (i) \( \angle BOE \) (ii) \( \angle OED \) (iii) \( \angle BFE \) ### Solution: #### (i) \( \angle BOE \) Since \( BC \) is a tangent to the circle at \( B \), \( \angle OBE \) is the angle between the radius \( OB \) and the tangent \( BC \). This angle is given as \( 20^\circ \). The angle subtended by the same arc at the center of the circle is twice the angle subtended at the circumference. Therefore, \( \angle BOE \) is twice \( \angle OBE \). \[ \angle BOE = 2 \times \angle OBE = 2 \times 20^\circ = 40^\circ \] #### (ii) \( \angle OED \) \( \angle OED \) is an angle at the center of the circle. To find this angle, we need to consider the angles around point \( O \). Since \( \angle BOE = 40^\circ \) and \( \angle CBD = 42^\circ \), we need to find the relationship between these angles. Notice that \( \angle OED \) is the external angle for triangle \( OBD \), and it is equal to the sum of the opposite internal angles. \[ \angle OED = \angle OBE + \angle CBD = 20^\circ + 42^\circ = 62^\circ \] #### (iii) \( \angle BFE \) To find \( \angle BFE \), we need to use the fact that \( \angle BFE \) is an angle in the alternate segment. This means it is equal to the angle subtended by the same arc at the circumference. Since \( \angle CBD = 42^\circ \), and \( \angle BFE \) is subtended by the same arc \( BD \), we have: \[ \angle BFE = \angle CBD = 42^\circ \] ### Summary of Answers: (i) \( \angle BOE = 40^\circ \) (2 marks) (ii) \( \angle OED = 62^\circ \) (2 marks) (iii) \( \angle BFE = 42^\circ \) (3 marks)