1. **Problem statement:** Find the unknown angles in each quadrilateral or polygon given the known angles. 2. **Key formula:** The sum of interior angles of a quadrilateral is $$360^\circ$$. For a triangle, the sum of interior angles is $$180^\circ$$. 3. **Part a:** Given angles: $$90^\circ, 70^\circ, 105^\circ, x^\circ$$. Sum of angles in quadrilateral: $$90 + 70 + 105 + x = 360$$. Calculate $$x$$: $$x = 360 - (90 + 70 + 105) = 360 - 265 = 95^\circ$$. 4. **Part b:** Given angles: $$90^\circ, 2x^\circ, 105^\circ, x^\circ$$. Sum: $$90 + 2x + 105 + x = 360$$ Simplify: $$195 + 3x = 360$$ Solve for $$x$$: $$3x = 360 - 195 = 165$$ $$x = \frac{165}{3} = 55^\circ$$. 5. **Part c:** Given angles: $$3a^\circ, 108^\circ, 2a^\circ, a^\circ$$. Sum: $$3a + 108 + 2a + a = 360$$ Combine like terms: $$6a + 108 = 360$$ Solve for $$a$$: $$6a = 360 - 108 = 252$$ $$a = \frac{252}{6} = 42^\circ$$. Calculate each angle: $$3a = 126^\circ, 2a = 84^\circ, a = 42^\circ$$. 6. **Part d:** Given a triangle with angles $$140^\circ, 88^\circ, a^\circ, b^\circ, c^\circ$$ and an external angle $$80^\circ$$. Since a triangle has only 3 angles, likely a, b, c are unknown angles of a polygon or additional angles. Assuming a quadrilateral with angles $$140^\circ, 88^\circ, a^\circ, b^\circ$$ and external angle $$80^\circ$$ adjacent to one side. Sum of quadrilateral angles: $$140 + 88 + a + b = 360$$ External angle $$80^\circ$$ relates to internal angle adjacent to it: $$c = 180 - 80 = 100^\circ$$ (if c is internal angle adjacent to external 80°). Then sum: $$140 + 88 + a + b + c = 360 + 100 = 460$$ which is inconsistent. Without more info, cannot solve uniquely. 7. **Part e:** Given angles $$90^\circ, y^\circ, x^\circ, z^\circ, w^\circ, 130^\circ$$. Sum of angles in a polygon with 6 sides (hexagon) is: $$ (6-2) \times 180 = 720^\circ$$. Sum known: $$90 + 130 = 220$$ Sum unknown: $$x + y + z + w = 720 - 220 = 500^\circ$$. Cannot solve uniquely without more equations. 8. **Part f:** Given angles $$x^\circ, y^\circ, w^\circ, z^\circ, 45^\circ$$. Assuming pentagon, sum of interior angles: $$ (5-2) \times 180 = 540^\circ$$. Sum known: $$45^\circ$$ Sum unknown: $$x + y + w + z = 540 - 45 = 495^\circ$$. Cannot solve uniquely without more info. **Final answers:** - a) $$x = 95^\circ$$ - b) $$x = 55^\circ$$ - c) $$a = 42^\circ$$, angles: $$126^\circ, 108^\circ, 84^\circ, 42^\circ$$ - d), e), f) require more information to solve uniquely.