Final answer:
To find the missing angle measures, one must understand that the sum of the angles in a triangle is always 180°, while for a trapezoid and other polygons, different rules apply. Using these principles, the missing angles were calculated for the geometry problems presented.
Explanation:
The student is attempting to solve problems related to the measurements of angles in different geometrical figures: a triangle, a trapezoid, and a polygon with six angles.
Finding the missing angle in a triangle
Triangles have three angles that sum to 180°. Given two angles, 62° and 81°, we find the third by subtracting the sum of the known angles from 180°. Therefore, the third angle is 180° - 62° - 81° = 37°.
Finding the missing angles in a trapezoid
For a trapezoid with angles 2h°, 2h°, h°, and h°, if the trapezoid is isosceles, the opposite angles are equal. Thus we can set up the equation h + 2h = 90° to find h, deducing that h = 30°.
Finding the missing angles in a polygon
For a six-angled polygon, the sum of the angles should be (6-2)×180° = 720°. With the provided angles, we add up the known ones and subtract from 720 to find the two missing angles:
720° - 135° - 62° - 140° - 151° = 232°; then n + (n+6) = 232°, which gives 2n + 6 = 232° and n = 113°.