https://file.aiquickdraw.com/mathbot/file/1736974301943c18jpin0.jpg

### Solution:

**a) Find the sizes of angles \(x\), \(y\), and \(z\):**

1. **Angle \(x\):**
   - The angles around point \(C\) must sum to \(360^\circ\). These angles are \(97^\circ\), \(37^\circ\), and \(x\).
   - Using the straight-line property at \(C\), the two angles \(x\) and \(46^\circ\) form a supplementary pair:
     \[
     x + 46^\circ = 180^\circ
     \]
     \[
     x = 180^\circ - 46^\circ = 134^\circ
     \]

2. **Angle \(y\):**
   - Triangle \(DEC\) has one angle given (\(97^\circ\)) and one angle (\(x\)) found in step 1.
   - The sum of the angles in a triangle is \(180^\circ\):
     \[
     y + 97^\circ + 134^\circ = 180^\circ
     \]
     \[
     y = 180^\circ - (97^\circ + 134^\circ)
     \]
     \[
     y = 49^\circ
     \]

3. **Angle \(z\):**
   - Triangle \(DEC\) has angles \(y = 49^\circ\), \(97^\circ\), and \(z\).
   - Using the triangle angle sum property:
     \[
     z + 97^\circ + y = 180^\circ
     \]
     \[
     z = 180^\circ - (97^\circ + 49^\circ)
     \]
     \[
     z = 34^\circ
     \]

---

**b) Why are triangles \(ABC\) and \(DEC\) similar?**

Triangles \(ABC\) and \(DEC\) are similar because they have the same angle measures:
- Angle \(A = \angle E = 37^\circ\),
- Angle \(B = \angle D = 46^\circ\),
- Angle \(C = \angle C = 97^\circ\) (common angle).

By the **Angle-Angle-Angle (AAA)** similarity criterion, triangles \(ABC\) and \(DEC\) are similar.

---

Do you need additional steps explained or clarifications? 😊

### Related Questions:
1. What is the triangle angle sum property, and how is it applied?
2. What are the criteria for two triangles to be similar?
3. Can \(x\), \(y\), or \(z\) change if the figure is not to scale? Why not?
4. How do supplementary angles around a point work in geometry?
5. Why does the common angle guarantee similarity between triangles?

### Tip:
Always check if two triangles share a common angle when determining similarity—it simplifies the reasoning process!

Work out the sizes of angles x, y, and z below. Write a sentence to explain how you know that triangle ABC is similar to triangle DEC.

Discover how to calculate the sizes of angles x, y, and z in a geometric problem involving triangle ABC and triangle DEC. Learn about triangle similarity using the AAA criterion and key angle properties. Suitable for students in Grades 8-10.

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