To find the unknown angles $x$ and $y$, we will use basic angle relationships in triangles.

Step 1: Identify the known angles in the triangle.

• One angle is given as $40^\circ$,
• Another is $51^\circ$,
• Another angle is $76^\circ$.

Step 2: Use the property that the sum of the angles in a triangle is always $180^\circ$.

Let’s focus on the triangle with angles $40^\circ$, $51^\circ$, and the unknown angle $y$.

$40^\circ + 51^\circ + y^\circ = 180^\circ$ $91^\circ + y^\circ = 180^\circ$ $y = 180^\circ - 91^\circ = 89^\circ$

Step 3: Use the straight line angle property.

On a straight line, the sum of the adjacent angles must be $180^\circ$. Since $y$ and $x$ are adjacent on a straight line, we can use:

$x^\circ + y^\circ = 180^\circ$ $x^\circ + 89^\circ = 180^\circ$ $x = 180^\circ - 89^\circ = 91^\circ$

Final answers:

$x = 91^\circ$ $y = 89^\circ$

Would you like further details on the reasoning? Here are some additional related questions:

1. How does the sum of angles in any triangle always total to 180°?
2. Can adjacent angles on a straight line sum to more or less than 180°?
3. How would this solution change if the triangle were non-Euclidean?
4. What is the difference between interior and exterior angles in polygons?
5. Can we use similar triangle properties to solve for angles?

Tip: Always remember that adjacent angles on a straight line add up to 180°. This is useful for quickly finding unknown angles in geometrical problems.