1. **Find the value of $n$ in the triangle with angles 112°, 100°, and $n$.** The sum of angles in any triangle is $180^\circ$. $$112 + 100 + n = 180$$ $$212 + n = 180$$ $$n = 180 - 212 = -32\,\text{degrees}$$ Since this is not possible for a triangle, likely the figure is a cyclic quadrilateral or there is an error. But interpreting as triangle, angle sum must be $180^\circ$. 2. **Find values of $q, r, s$ in a polygon inside the circle with angles 31°, 61°, $q, r, s$.** Since they are angles in a polygon, typically sum of interior angles for a polygon with $n$ sides is $(n-2) \times 180^\circ$. Not enough info to find all three variables directly. But assuming quadrilateral (4 sides) inside circle (cyclic), sum of opposite angles is $180^\circ$. If $31^\circ$ and $q$ are opposite angles, then: $$31 + q = 180 \Rightarrow q = 149^\circ$$ If $61^\circ$ and $r$ are opposite angles: $$61 + r = 180 \Rightarrow r = 119^\circ$$ Then $s$ would be the remaining angle in the quadrilateral: Sum of all angles = $360^\circ$ $$31 + 61 + 149 + 119 = 360$$ $$360 = 360$$ So $s$ is not independent or may equal one of the above or the question might have extra variable. Without more info, assume $s$ completes the figure. 3. **Find $z$ in the polygon with angles 50°, 100°, $z$, and 40°.** Sum of angles in polygon with 4 sides (quadrilateral) is $360^\circ$. $$50 + 100 + z + 40 = 360$$ $$190 + z = 360$$ $$z = 360 - 190 = 170^\circ$$ 4. **Find $q$ in triangle with angle 152° (inside polygon) and unknown $q$.** Sum of angles in triangle is $180^\circ$ $$152 + q + x = 180$$ Without $x$, cannot find $q$ exactly. Assuming $x$ is the other interior angle, then: $q$ is an exterior angle, so: $$q = 152 + x$$ Or if $q$ is the other interior angle: Not enough info. 5. **Complete the table for regular polygons with $n = 6,7,8,9,10,12$ sides.** Name and sum of interior angles can be computed: - Number of sides $n$. - Name is standard polygon names. - Angle sum of polygon = $180(n-2)$ degrees. | Number of sides | Name | Angle Sum | |-----------------|-----------------|-------------------| | 5 | Pentagon | 540 degrees | | 6 | Hexagon | $180(6-2)=720$ | | 7 | Heptagon | $180(7-2)=900$ | | 8 | Octagon | $180(8-2)=1080$ | | 9 | Nonagon | $180(9-2)=1260$ | | 10 | Decagon | $180(10-2)=1440$ | | 12 | Dodecagon | $180(12-2)=1800$ | **Final answers:** - Part a) $n = -32^\circ$ (indicates a problem with figure or type) - Part b) $q=149^\circ$, $r=119^\circ$, $s$ determined by polygon context - Part c) $z=170^\circ$ - Part d) Not enough info to find $q$ - Table completed with polygon names and angle sums.