The diagram shows two missing angles, $x$ and $y$, along with a given angle of $64^\circ$. Let's break this down to find the missing angles.

Step-by-step solution:

1. Identify angle $x$:

   • The angle $x$ and the given angle of $64^\circ$ are supplementary, meaning they form a straight line. Therefore, the sum of these two angles is $180^\circ$. $x + 64^\circ = 180^\circ$ Solving for $x$: $x = 180^\circ - 64^\circ = 116^\circ$

2. Identify angle $y$:

   • The angle $y$ is complementary to the given right angle (since it forms part of the right-angled corner in the diagram). The sum of complementary angles is $90^\circ$. $y + 64^\circ = 90^\circ$ Solving for $y$: $y = 90^\circ - 64^\circ = 26^\circ$

Final answers:

• $x = 116^\circ$
• $y = 26^\circ$

Let me know if you want further clarifications or details!

Here are some related questions to expand your understanding:

1. What are supplementary angles?
2. How do complementary angles relate to right angles?
3. Can the value of $x$ change if the given angle was not 64°?
4. How do we identify which angles are supplementary or complementary in a given diagram?
5. What are vertical angles and how are they calculated?

Tip: Always check if angles form a straight line (supplementary) or a right angle (complementary) to simplify your calculations.