Why is the prayer carpet at an angle in some mosques?
The carpet is at an angle because the mosque building doesn't face the qibla, but the prayer rows must. Many Western mosques are converted shops, churches, houses or warehouses built along the street, not toward Mecca. So the carpet lines are laid diagonally to point everyone toward the Kaaba.
The rows follow the qibla, not the walls
Facing the Kaaba is a condition of the obligatory prayer. Purpose-built mosques are usually designed so the front wall faces the qibla, and the carpet runs straight. But in Europe and the USA, many communities buy whatever building they can afford. A terraced house in Birmingham or a former warehouse in Chicago won't happen to line up with Mecca, so the carpet, or the tape lines on it, run at an angle across the room.
For the basics on the direction itself, see What is the qibla?
Why the direction can look "wrong"
From North America, the qibla points roughly northeast, not southeast as many people expect from a flat map. That's because the shortest path on a globe follows a great circle. From the UK it's roughly southeast. When a mosque re-measures with modern tools, it sometimes finds the old carpet was a few degrees off and re-lays it at a new angle. See How do I find the qibla without an app?
Is a small error a problem?
No. Most scholars say that when you can't see the Kaaba, facing its general direction is enough, and a small deviation doesn't spoil the prayer. If you prayed for years in a slightly-off direction in good faith, those prayers are valid. Still, communities make an effort to get it right, and today it's easy to check with a compass or app. See Can I use a qibla app to find the prayer direction?
Straight rows matter too
The angled lines also help people stand in straight, tight rows. The Prophet ﷺ said, "Straighten your rows, for straightening the rows is part of establishing the prayer" (Sahih al-Bukhari 723). Patterned prayer carpets with one arch per person make this even easier: stand on the line, toes behind it, shoulder to shoulder with the person next to you.
- Rows must face the qibla, even if the building doesn't.
- Converted Western buildings often need angled carpets.
- Facing the general direction is enough; small errors are forgiven.
- The lines also help keep the rows straight.
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