Sharing endpoint at infinity

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I could not solve this problem. Can you help me please?

Prove that if m and n are hyperbolic lines that share an endpoint at infinity, then there does not exist a hyperbolic line perpendicular to both m and n.

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If such a line existed, you'd get an ideal triangle with angle sum 180 degrees.

But every hyperbolic triangle has angle sum strictly less than 180 degrees.