To determine the values of angles $x^\circ$ and $y^\circ$, we can use properties of parallel lines and angles in a triangle.

Step-by-step Solution:

1. Using the fact that the top and bottom lines are parallel:

• The two angles of $55^\circ$ are alternate interior angles, because the top and bottom lines are parallel, and the lines creating the angles act like transversals.
• This means that the triangles on either side of the middle triangle are isosceles, and each base angle is $55^\circ$.

2. Find $x^\circ$:

• In the middle triangle, we know one angle is $55^\circ$. Since the angles in a triangle sum to $180^\circ$, we can find $x$ by subtracting the known angles: $x^\circ = 180^\circ - 55^\circ - 55^\circ = 70^\circ$

3. Find $y^\circ$:

• On the right side of the diagram, we have a triangle where one angle is $55^\circ$ and the interior angle $y^\circ$ is adjacent to it. Since these two angles form a straight line (sum to $180^\circ$), we can find $y$ as: $y^\circ = 180^\circ - 55^\circ = 125^\circ$

Final Answers:

• $x = 70^\circ$
• $y = 125^\circ$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the sum of interior angles in any triangle?
2. How do alternate interior angles help determine parallel lines?
3. How do you calculate the third angle in a triangle if two angles are known?
4. What is the relationship between adjacent angles on a straight line?
5. How do isosceles triangles help solve for missing angles?

Tip: Remember that when two lines are parallel, alternate interior angles are always congruent.