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In the provided image, problem 9 asks to calculate the angles ∠Y and ∠Z in the triangle XYZ, given the following information:

- \( XZ = YZ \) (indicating triangle XYZ is isosceles)
- \( \angle X = 56^\circ \)
- \( \angle ZYM = 40^\circ \) (since YZM is a straight line).

### Step-by-step solution:

1. **Isosceles Triangle Property**: Since \( XZ = YZ \), triangle XYZ is isosceles. Therefore, \( \angle Y = \angle Z \) (the base angles are equal).

2. **Angle at Z**: 
   Since \( YZM \) is a straight line, the sum of the angles on this line must be \( 180^\circ \). So:
   \[
   \angle ZYM + \angle Z = 180^\circ
   \]
   Substituting \( \angle ZYM = 40^\circ \), we get:
   \[
   40^\circ + \angle Z = 180^\circ
   \]
   Solving for \( \angle Z \):
   \[
   \angle Z = 180^\circ - 40^\circ = 140^\circ
   \]

3. **Angle Sum in Triangle XYZ**: The sum of the interior angles in a triangle is always \( 180^\circ \). For triangle XYZ, we know \( \angle X = 56^\circ \), and now \( \angle Z = \angle Y \). So:
   \[
   \angle X + 2 \times \angle Z = 180^\circ
   \]
   Substituting \( \angle X = 56^\circ \):
   \[
   56^\circ + 2 \times \angle Z = 180^\circ
   \]
   Solving for \( \angle Z \):
   \[
   2 \times \angle Z = 180^\circ - 56^\circ = 124^\circ
   \]
   \[
   \angle Z = \frac{124^\circ}{2} = 62^\circ
   \]

Thus, the values of the angles are:

- \( \angle Z = 62^\circ \)
- \( \angle Y = 62^\circ \)

Would you like any additional details on this solution or have any further questions?

### Follow-up Questions:
1. What is the sum of interior angles in any triangle?
2. How does the property of isosceles triangles help in solving for angles?
3. How would the solution change if \( XZ \neq YZ \)?
4. What is the rule for angles along a straight line?
5. Can you apply the angle sum property to a quadrilateral?

### Tip:
When working with isosceles triangles, remember that the two base angles are always equal. This often simplifies calculations when given one or more angles.

Calculate ∠Y and ∠Z in triangle XYZ given that XZ = YZ and YZM is a straight line. ∠X = 56° and ∠ZYM = 40°.

Learn how to solve for ∠Y and ∠Z in an isosceles triangle XYZ where XZ = YZ, given ∠X = 56° and ∠ZYM = 40°. This problem involves the angle sum property, the isosceles triangle theorem, and the straight line angle rule. Suitable for Grades 7-10.

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