1. **Problem statement:** Find the unknown angle measures indicated by question marks in four different circle and tangent line configurations. 2. **Key formulas and rules:** - The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. - The sum of angles in a triangle is 180°. - Angles around a point sum to 360°. - Tangent lines are perpendicular to the radius at the point of tangency. 3. **Solution for each problem:** **13) Given angle inside the circle is 63°, find the outside angle at the apex of the triangle.** - The sum of angles around the point outside the circle is 360°. - Given other angles are 90°, 40°, and 117°. - Calculate unknown angle $x$: $$x = 360 - 90 - 40 - 117 = 113°$$ **14) Given outside angle 44°, find inside angle at the circle.** - Angle between tangent and chord equals angle in alternate segment. - Therefore, inside angle = 44°. **15) Given inside angle 117°, find outside angle at apex.** - Tangent is perpendicular to radius, so one angle is 90°. - Triangle angles sum to 180°. - Calculate unknown angle $x$: $$x = 180 - 90 - 117 = -27°$$ - Negative angle is impossible, so re-check: likely the outside angle is supplementary to inside angle. - Outside angle = $180 - 117 = 63°$. **16) Given outside angle 52°, find inside angle between chord and tangent.** - Angle between tangent and chord equals angle in alternate segment. - Therefore, inside angle = 52°. 4. **Final answers:** - 13) $113°$ - 14) $44°$ - 15) $63°$ - 16) $52°$