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To calculate the size of angle \( \angle ABC \) in the given triangle, we can use the **Law of Cosines** because we have two sides of the triangle and an included angle. The Law of Cosines is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( c \) is the side opposite the angle we want to calculate, which in this case is \( BC = 20 \, \text{cm} \).
- \( a = AB = 18 \, \text{cm} \)
- \( b = AC = 20 \, \text{cm} \)
- \( \angle A = 60^\circ \) (given)

We can rearrange the Law of Cosines to solve for \( \cos(\angle ABC) \):

\[
\cos(\angle ABC) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Substitute the values:

\[
\cos(\angle ABC) = \frac{18^2 + 20^2 - 20^2}{2 \cdot 18 \cdot 20}
\]

Simplify:

\[
\cos(\angle ABC) = \frac{324 + 400 - 400}{2 \cdot 18 \cdot 20} = \frac{324}{720} = 0.45
\]

Now, find the angle using the inverse cosine:

\[
\angle ABC = \cos^{-1}(0.45) \approx 63.26^\circ
\]

Thus, \( \angle ABC \approx 63.26^\circ \).

Would you like further details or clarifications on this solution?

Here are five related questions you might find interesting:
1. How can we verify the solution using another method, such as the Law of Sines?
2. How would you calculate the third angle \( \angle BCA \) of the triangle?
3. How does the Law of Cosines apply when the angle is obtuse?
4. What are the common uses of the Law of Cosines in real-world applications?
5. How does the Law of Cosines relate to the Pythagorean theorem?

**Tip**: The Law of Cosines is especially useful when you don't have a right triangle, but you know two sides and the included angle, or all three sides of a triangle.

Solve a geometry problem involving the calculation of angle ABC in a triangle. This problem uses the Law of Cosines with given sides of 18 cm and 20 cm, and an included angle of 60°. Learn step-by-step how to use trigonometry to find angle ABC, which is approximately 63.26°.

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