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.
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°.
