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We are given a geometric diagram showing two triangles, \(\triangle ABC\) and \(\triangle CDE\), sharing a common point \(C\), and the values of several angles are marked on the diagram. The goal is to find the value of angle \(y\) in triangle \(\triangle ABC\).

Here is the breakdown of how to solve for \(y\):

### Step-by-Step Solution:

1. **Determine angle \(C\) (at point \(C\)) in \(\triangle ABC\):**
   - The angle given at point \(C\) is \(72^\circ\). This angle is adjacent to both triangles \(\triangle ABC\) and \(\triangle CDE\).

2. **Use the exterior angle property:**
   - At point \(E\), an exterior angle of \(\triangle CDE\) is given as \(124^\circ\). By the exterior angle theorem:
     \[
     \text{Exterior Angle} = \text{Sum of the two opposite interior angles}.
     \]
     Hence,
     \[
     \angle DCE + \angle CED = 124^\circ.
     \]
     Since \(\angle DCE = 72^\circ\), we can solve for \(\angle CED\):
     \[
     \angle CED = 124^\circ - 72^\circ = 52^\circ.
     \]

3. **Use the sum of angles in a triangle:**
   - In \(\triangle ABC\), the sum of the angles must equal \(180^\circ\). Therefore:
     \[
     \angle A + \angle B + \angle C = 180^\circ.
     \]
     We know \(\angle C = 72^\circ\) (given), and we are solving for \(\angle B = y\). Let's denote \(\angle A = x\).

4. **Determine angle \(x\) (at point \(A\)):**
   - From the diagram, observe that angle \(x\) is a linear pair with angle \(72^\circ\) at point \(C\). Since the sum of linear pair angles is \(180^\circ\), we have:
     \[
     x + 72^\circ = 180^\circ \quad \Rightarrow \quad x = 180^\circ - 72^\circ = 108^\circ.
     \]

5. **Solve for angle \(y\):**
   - Now, in \(\triangle ABC\), using the sum of angles:
     \[
     x + y + 72^\circ = 180^\circ,
     \]
     Substituting \(x = 108^\circ\):
     \[
     108^\circ + y + 72^\circ = 180^\circ,
     \]
     Simplifying:
     \[
     y + 180^\circ = 180^\circ,
     \]
     \[
     y = 0^\circ.
     \]

Thus, the value of angle \(y\) is \(0^\circ\), which seems incorrect, I will correct

Solve a geometric problem involving two triangles and angle properties to find the value of angle y. Learn how the sum of angles in a triangle and the exterior angle theorem apply in this problem. Suitable for students in Grades 8-10.

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