In a circle with center O, XT and ZT are tangents to the circle at X and Z. If angle OZX = 25 degrees, calculate angle OXZ, angle XOZ, and angle XWZ.

Step 1: Identify given information and make an assumption. Given that XT and ZT are tangents to a circle with center O at points X and Z respectively. The diagram shows OZX = 25^. The text states OZY = 25^. We will assume that OZY in the text refers to OZX as depicted in the diagram. i) Calculate OXZ OX and OZ are radii of the same circle, so OX = OZ. Therefore, OXZ is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. OXZ = OZX From the diagram, OZX = 25^. OXZ = 25^ Reason: Base angles of an isosceles triangle are equal. ii) Calculate XOZ The sum of angles in a triangle is 180^. In OXZ: XOZ + OXZ + OZX = 180^ Substitute the known values: XOZ + 25^ + 25^ = 180^ XOZ + 50^ = 180^ XOZ = 180^ - 50^ XOZ = 130^ Reason: Sum of angles in a triangle is 180^. iii) Calculate XWZ XWZ is the angle subtended by arc XZ at the circumference. XOZ is the angle subtended by the same arc XZ at the center. The angle at the center is twice the angle at the circumference subtended by the same arc. XWZ = (1)/(2) XOZ Substitute the value of XOZ: XWZ = (1)/(2) × 130^ XWZ = 65^ Reason: Angle at the center is twice the angle at the circumference. iv) Calculate XTZ OX is a radius and XT is a tangent at point X. A radius is perpendicular to the tangent at the point of tangency. OXT = 90^ Similarly, OZ is a radius and ZT is a tangent at point Z. OZT = 90^ Consider the quadrilateral OXTZ. The sum of interior angles in a quadrilateral is 360^. XTZ + OXT + OZT + XOZ = 360^ Substitute the known values: XTZ + 90^ + 90^ + 130^ = 360^ XTZ + 310^ = 360^ XTZ = 360^ - 310^ XTZ = 50^ Reason: Radius is perpendicular to tangent at the point of contact. Sum of angles in a quadrilateral is 360^. Final Answers: i) 25^ ii) 130^ iii) 65^ iv) 50^ Send me the next one 📸