1. **Stating the problem:** We are given a polygon with angles labeled as 42°, y°, 40°, 92°, 153°, and x°. We need to find the values of the unknown angles $x$ and $y$. 2. **Formula used:** The sum of interior angles of a polygon with $n$ sides is given by: $$\text{Sum of interior angles} = (n-2) \times 180^\circ$$ 3. **Determine the number of sides:** The polygon has points A, B, C, D, and E, so it is a pentagon with $n=5$ sides. 4. **Calculate the sum of interior angles:** $$ (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ $$ 5. **Sum of known and unknown angles:** The interior angles are $42^\circ$, $y^\circ$, $40^\circ$, $92^\circ$, $153^\circ$, and $x^\circ$. Since the polygon has 5 angles, and the problem shows 6 angles, we must identify which angles are interior. Assuming $42^\circ$ and $y^\circ$ are at the same vertex (A), their sum is the interior angle at A. Similarly, $92^\circ$ and $153^\circ$ are at vertex D, so their sum is the interior angle at D. 6. **Calculate interior angles at vertices A and D:** $$ \text{Angle at A} = 42^\circ + y^\circ $$ $$ \text{Angle at D} = 92^\circ + 153^\circ = 245^\circ $$ 7. **Sum of all interior angles:** $$ (42 + y) + 40 + x + 245 + \text{angle at E} = 540 $$ 8. **Find angle at E:** Since the polygon has 5 vertices, the angles are at A, B, C, D, and E. The angles at B and C are given as $40^\circ$ and $x^\circ$ respectively. So the sum is: $$ (42 + y) + 40 + x + 245 + \text{angle at E} = 540 $$ 9. **Calculate angle at E:** $$ \text{angle at E} = 540 - (42 + y + 40 + x + 245) = 540 - (327 + x + y) = 213 - x - y $$ 10. **Use the fact that angle at E is an interior angle and must be positive and less than 180°:** $$ 0 < 213 - x - y < 180 $$ 11. **Use the polygon's exterior angle properties or additional information to solve for $x$ and $y$. Since the problem does not provide more data, we assume the polygon is simple and convex, so the sum of angles at A and D must be less than 360°. 12. **Solve for $y$ using the straight angle at A:** Since $42^\circ$ and $y^\circ$ are adjacent angles at vertex A, and assuming they form a straight line: $$ 42 + y = 180 $$ $$ y = 180 - 42 = 138 $$ 13. **Solve for $x$ using the straight angle at C:** Assuming $x^\circ$ is the interior angle at C, and the adjacent angle is $40^\circ$ at B, and the polygon is convex, then: $$ x = 180 - 40 = 140 $$ 14. **Final answers:** $$ x = 140^\circ $$ $$ y = 138^\circ $$