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The problem involves a regular octagon, and you're tasked with showing that the angle \( \angle ZAB \) is 67.5°. Let's break it down:

### Step 1: Understanding the Diagram
- \( AZ \) is given as 120 cm, where \( Z \) is the center of the regular octagon.
- The shape is a regular octagon, meaning all sides and interior angles are equal.
- We are asked to find \( \angle ZAB \), which is the angle subtended at point \( A \) between lines \( ZB \) and \( AZ \).

### Step 2: Central Angle Calculation
In a regular octagon:
- The octagon has 8 equal sides.
- The total sum of the angles around point \( Z \) (the center of the octagon) is 360°.
- Each central angle subtended by two adjacent vertices of the octagon (like \( ZAB \)) is given by:
  \[
  \text{Central angle} = \frac{360^\circ}{8} = 45^\circ
  \]
So, \( \angle AZB = 45^\circ \).

### Step 3: Using Isosceles Triangle Properties
- Triangle \( AZB \) is isosceles since \( AZ = BZ \) (radii of the octagon).
- In an isosceles triangle, the two base angles are equal. Let’s call these angles \( \theta \).
- The sum of angles in a triangle is always 180°, so for triangle \( AZB \):
  \[
  45^\circ + 2\theta = 180^\circ
  \]
  Solving for \( \theta \):
  \[
  2\theta = 180^\circ - 45^\circ = 135^\circ
  \]
  \[
  \theta = \frac{135^\circ}{2} = 67.5^\circ
  \]
Thus, \( \angle ZAB = 67.5^\circ \), as required.

Would you like further clarification on this, or details about the geometry of regular polygons?

### Related Questions:
1. What is the sum of all interior angles of a regular octagon?
2. How would the calculation change if the shape was a hexagon instead of an octagon?
3. How can you calculate the exterior angles of a regular polygon?
4. What is the area of the octagon if \( AZ = 120 \, \text{cm} \)?
5. How do the properties of regular polygons help in constructing such shapes?

### Tip:
For regular polygons, knowing the relationship between the number of sides and central angles can help solve many geometric problems efficiently.

Show that the size of angle ZAB is 67.5°.

Learn how to show that angle ZAB in a regular octagon is 67.5°. This solution breaks down geometric concepts such as central angles, isosceles triangles, and theorems related to regular polygons. Suitable for Grades 10-12 geometry students.

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