Tangent line angle relationships in a circle problem

Expert intro: A tangent to a circle gives you one very reliable fact: it is perpendicular to the radius at the point of tangency. After that, the rest depends on the marked angles in the diagram.

Detailed walkthrough

I can’t confirm which choice is incorrect without the diagram, because the angle labels $1$–$4$ depend on the exact layout.

What the diagram is usually testing:

• A tangent is perpendicular to the radius at the point of contact, so that angle is $90^\circ$.
• Angles on a straight line add to $180^\circ$.
• If two angles are a linear pair, one can be found by subtracting from $180^\circ$.
• If two lines cross, vertical angles are equal.

So the statement that is “not true” is the one that conflicts with one of those facts. If the diagram shows $\ell$ tangent at $P$, then any angle formed by the tangent and radius at $P$ must be right, not acute or obtuse.

If you share the figure, I can tell you exactly whether A, B, C, or D is the wrong statement.

💡 Pitfall guide

Don’t assume every angle near a tangent is $90^\circ$. Only the angle between the tangent and the radius at the point of tangency is guaranteed to be right. Angles formed elsewhere in the circle can be very different.

🔄 Real-world variant

If the tangent is paired with a chord, the tangent-chord angle theorem may apply: the angle between a tangent and a chord equals half the measure of the intercepted arc. That gives a different route than the radius-perpendicular rule.

🔍 Related terms

tangent to a circle, radius, point of tangency