The image below shows a Bluetooth mini-speaker - Leaving Cert DCG - Question A-1 - 2017

To find the true angle between the lines ab and cd, follow these steps:

1. Identify Coordinates: Begin by establishing the coordinates of points a, b, c, and d based on the axonometric projection. Let’s denote them as follows:

   • a(x1, y1, z1)
   • b(x2, y2, z2)
   • c(x3, y3, z3)
   • d(x4, y4, z4)

2. Vector Representation: Construct vectors based on the identified points:

   • Vector AB = B - A = (x2 - x1, y2 - y1, z2 - z1)
   • Vector CD = D - C = (x4 - x3, y4 - y3, z4 - z3)

3. Dot Product: Calculate the dot product of the two vectors: ext{Dot Product} = AB ullet CD = (x2 - x1)(x4 - x3) + (y2 - y1)(y4 - y3) + (z2 - z1)(z4 - z3)

4. Magnitude of Vectors: Calculate the magnitudes of both vectors: $|AB| = ext{sqrt}((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)$ $|CD| = ext{sqrt}((x4 - x3)^2 + (y4 - y3)^2 + (z4 - z3)^2)$

5. Angle Calculation: Use the dot product to find the cosine of the angle θ between the two vectors: ext{cos}( heta) = rac{AB ullet CD}{|AB| |CD|} Finally, calculate the angle θ using the inverse cosine function: heta = ext{arccos}igg(rac{AB ullet CD}{|AB| |CD|}igg)

This will give you the true angle between the lines ab and cd.