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.