To solve for the values of $x$ and $y$ in the given diagram, let’s analyze the geometry step by step.

Step 1: Understanding the Diagram

• The triangle on the left contains angles $41^\circ$, $78^\circ$, and $y^\circ$.
• The two triangles share a common base and appear to form a straight line, which means the angles $y^\circ$ and $x^\circ$ are supplementary (sum to $180^\circ$).
• Also, the tick marks on the triangle sides indicate isosceles triangles, which means base angles are equal.

Step 2: Finding $y$ (Left Triangle)

The angles of a triangle add up to $180^\circ$. Therefore, in the left triangle: $41^\circ + 78^\circ + y = 180^\circ$ Solve for $y$: $y = 180^\circ - 41^\circ - 78^\circ$ $y = 61^\circ$

Step 3: Finding $x$ (Supplementary Angles)

The angle $x$ and $y$ are supplementary because they form a straight line: $x + y = 180^\circ$ Substitute $y = 61^\circ$ into the equation: $x + 61^\circ = 180^\circ$ $x = 180^\circ - 61^\circ$ $x = 119^\circ$

Final Answers:

• $y = 61^\circ$
• $x = 119^\circ$

Verification:

1. The triangle angles add to $180^\circ$ (left triangle: $41^\circ, 78^\circ, 61^\circ$).
2. $x + y = 180^\circ$ confirms that the supplementary relationship holds.

Would you like a more detailed breakdown, or do you have further questions? Here are five related questions:

1. How do we determine if a triangle is isosceles from tick marks?
2. What is the triangle angle sum property?
3. How can we find supplementary angles in geometric diagrams?
4. What is the difference between interior and exterior angles?
5. How do we solve for missing angles in more complex polygons?

Tip: Always check if two angles form a straight line (add up to $180^\circ$) when solving for supplementary angles.