Two identical circles touch each other externally at point Z. XY is a direct common tangent, which touches the circles at X and Y respectively. What is the ∠XYZ ?

90 degree

Let P be a point on XY such that, PZ is at right angles to the Line Joining the centers of the circles.

Note that, PZ is a common tangent to both circles.

This is because tangent is perpendicular to radius at point of contact for any circle.

let ∠PXZ= α and ∠PYZ = β.

PX = PZ [lengths of the tangents from an external point Z]

In a triangle ZXP, ∠PXZ = ∠XZP = α

similarly PY = ZP and ∠PZY = ∠ZYP = β

now in the triangle XZB,

∠ZXY + ∠ZYX + ∠XZY = 180°   [sum of the interior angles in a triangle]

α + β + (α + β) = 180°   (Since ∠XZY = ∠XZP + ∠PZY = α + β.

2α + 2β = 180°

α + β = 90°

Therefore, ∠XZY = α + β = 90°