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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.

Find the unknown angle measures in the triangle given the angles 40°, 51°, and 76°.

Learn how to find the unknown angles in a triangle with given angles 40°, 51°, and 76°. The step-by-step solution explains the angle sum property and the concept of adjacent angles on a straight line. Suitable for students in Grades 6-8.

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