How do you solve angle and arc questions in a tangent circle diagram?

Key concept: Problems like this usually mix three tools: tangent-radius facts, inscribed angles, and arc relationships. Once you spot which angle belongs to which arc, the rest becomes a chain of substitutions.

Step by step

Step 1: Use the circle facts you know

Here are the rules that usually drive the whole problem:

• A tangent is perpendicular to the radius at the point of tangency.
• An inscribed angle equals half its intercepted arc.
• A central angle equals the measure of its intercepted arc.
• A diameter subtends a semicircle, so that arc is $180^\circ$.

Step 2: Start from the given measures

You are given:

• $\overset{\frown}{PQ}=115^\circ$
• $m\angle RPS=36^\circ$
• $QS$ is a diameter
• $VQ$ is tangent at $Q$

From $\angle RPS=36^\circ$, the intercepted arc $RS$ is:

$\overset{\frown}{RS}=2\cdot 36^\circ=72^\circ$

Step 3: Use the diameter

Since $QS$ is a diameter, each semicircle measures $180^\circ$. So the arc from $Q$ to $S$ through $P$ and $R$ must fit into that semicircle depending on the diagram order.

If the points are arranged on the same arc path as listed, then the remaining arcs are found by subtraction from $180^\circ$ or by using the full circle total of $360^\circ$.

Step 4: Use tangent and inscribed-angle facts

Because $VQ$ is tangent at $Q$:

$\angle PQV = \frac{1}{2}\overset{\frown}{PQ}$

so

$\angle PQV = \frac{115^\circ}{2}=57.5^\circ$

Also, the angle between tangent $QV$ and chord $QS$ is $90^\circ$ because a tangent is perpendicular to the radius, and $QS$ is a diameter through the center.

Step 5: Build the remaining angles from the diagram order

The exact values of the labeled angles $\angle R$, $\angle S$, $\angle QPR$, $\angle QPS$, $\angle QTP$, $\angle VQS$, and the arcs $\overset{\frown}{QR}$, $\overset{\frown}{PRQ}$, $\overset{\frown}{RSP}$ depend on the placement of the points in the missing diagram.

The reliable way to finish is:

1. convert any inscribed angle to its intercepted arc,
2. use the semicircle $180^\circ$ from $QS$,
3. use the full circle $360^\circ$ for arc totals,
4. use tangent-radius perpendicularity at $Q$.

So the one value we can determine immediately from the text is:

$\overset{\frown}{RS}=72^\circ$

and

$\angle PQV=57.5^\circ$

Pitfall alert

Do not treat a tangent like an ordinary secant. The tangent at $Q$ is perpendicular to the radius, and that fact is easy to miss. Another trap is assuming every angle label can be solved from the text alone when the missing diagram controls the arc order.

Try different conditions

If the diagram shows a different order of points on the circle, the arc subtraction changes even though the circle facts stay the same. The method does not change: inscribed angle means half the arc, central angle equals the arc, and a diameter always gives $180^\circ$ for a semicircle.

Further reading

inscribed angle, tangent line, intercepted arc