Let us solve the problem step by step.

Given Triangle Geometry Problem:

• We need to calculate $x^\circ$ and $y^\circ$ in the triangle.

Step 1: Use Triangle Sum Property

For a triangle, the sum of all interior angles is $180^\circ$. Looking at triangle $ABC$, we know: $\text{Angle } A + \text{Angle } B + \text{Angle } C = 180^\circ$

From the given diagram:

• Angle $A = x^\circ$
• Angle $B = 53^\circ$
• Angle $C = y^\circ$

Thus, we can write: $x + 53 + y = 180$ x + y = 127 \tag{1}

Step 2: Solve for $x$

Notice the small right triangle $\triangle ABD$ inside the figure:

• Angle $x + 90^\circ = 180^\circ$ (linear pair of angles). Thus: $x = 90^\circ$

Step 3: Solve for $y$

Using equation (1), substitute $x = 90$: $90 + y = 127$ $y = 37^\circ$

Final Answers:

• $x = 90^\circ$
• $y = 37^\circ$

Correct Choices:

• For $x$: C. 90
• For $y$: B. 37

Would you like more explanation on any step?

Follow-Up Questions:

1. What is the Triangle Sum Property and why does it hold true?
2. How can we verify the values of $x$ and $y$ using alternate methods?
3. How does the right-angle triangle $\triangle ABD$ influence the calculation of $x$?
4. If another angle in the diagram were given, how would it change the calculations?
5. Can the angles of a triangle ever sum to more or less than $180^\circ$?

Tip:

Always check for smaller triangles within complex diagrams. They often help simplify the problem by providing additional relationships!